Short of bringing in some variant of force (law enforcement, animal control, etc) or threat thereof, as several of the other answers suggest, you're going to need finesse and time.
Before you get started, I strongly recommend reading:
- "How to Win Friends and Influence People"
- "Influence; The Psychology of Persuasion"
- anything you can find about using the Socratic Method to ask questions that help people discover flaws in their own views (if you can look beyond the subject matter of religion and generalize the principles from examples, "A Manual for Creating Atheists" is a good one).
- anything you can find about how compassion and empathy can be trained or increased. There's scientific literature and probably books covering it - the general gist is that people are often very empathetic so long as you can prompt them to think about what things are like from your perspective, and usually the problem is that they just haven't thought to or don't understand your perspective enough to imagine it.
That's the bare minimum. Getting people whose actions and life decisions you disagree with to change their behavior to your liking is a very non-trivial science and art form. No answer will properly sum up the principles involved - because they take several books to properly cover.
However, here's an overall scaffold, that I present with some hypothetical examples as a necessarily crude general gist of an approach:
1 - Establish Friendliness and Understanding
Strike up a friendly/good-natured conversation with her, working quickly into something that shows her that you understand and can relate to her perspective. Maybe:
So I've noticed you feed the dogs regularly, you really care about them and their well-being, right?
If you've already given her a negative impression of you by challenging her on this topic before, start with something like:
Listen, I'm sorry I was harsh towards you before about the dogs. I understand that you care about these dogs and that you only feed them because you want to help them have better lives, right?
Most people need to trust that you understand and relate before they become receptive to opening themselves up to your criticisms. And people really react poorly to being told they can't do something - most people's minds abhor limits that they did not choose or agree with (the most likely time for people to rebel against authority? Immediately after a freedom they previously had is noticeably taken away).
Without a positive interpersonal foundation, "Stop doing [x]" is the worst of both: it implies you're wrong for wanting to do the thing, and tries to tell you you can't do that thing.
2 - Turn All Your Disagreements into Questions
Whenever I hear something wrong, I do my best to immediately ask myself in my head "how could this be right?". I look real hard for any way I can turn ever perceived error in reasoning, judgment, or knowledge, into a question I can ask them. For example:
Okay, now one part I don't full understand, that I'm worried about: You know how when you go into forests and stuff they tell you not to feed bears because then they start thinking of people as sources of food? I understand that dogs are different and friendlier to people than wild animals, so maybe I'm missing something here: don't you think there's a risk of that happening with dogs?
And if she says "no", you keep looking for questions. Ideally, you figure out why she thinks that's not an issue, or even better, what other emotional associations she has invested in the issue. Once you don't see a clear path any further, you can straight-up introduce data-points in question form:
Okay, I think I understand that [rephrase her reasoning while keeping seemingly significant words the same, then ask for confirmation]. Well, what about that kid that got bitten by one of the dogs the other day? You know these dogs better than I do, do you think that the kid did something wrong or might it be related?
Well, the reason I ask is because I've been noticing that whenever I go outside these dogs get closer to me, seemingly begging for food. I think they've been doing it more and more. Is it possible that the dogs are starting to expect a meal from humans in this apartment complex?
For best questions, don't follow these suggestions: Look inside yourself and really figure out what specific events or thoughts or experiences you have causing your concern about the dogs. Ask her about those. For example, maybe you have an experience in your past where a dog got used to people feeding it and then bit you when you were a kid, or something like that. Or maybe you've had a close call with one of these current street dogs biting you or one of your kids. Tell that story, in a way that describes but does not accuse, and instead of finishing with "and that's why you should stop" finish with "so that's why I am concerned - does that help make it clearer where I'm coming from?"
And perhaps better still, stop focusing on the problem you care about, and get her to identify a problem that she cares about:
What do you think will happen to these dogs once you've moved out of here? They seem so used to getting food from you, do you think they'll manage to find enough food in the wild on their own?
I've noticed that some of the other residents have been getting very hostile to the dogs - do you ever worry that they might start calling animal control services on them, or that one of the dogs might bite a kid and end up being put down?
Important: Most of these questions can sound disingenuous or passive-aggressive unless you've actually established yourself as being there to ask genuine questions. The last one can even come off as a "would be a shame if someone were to... kill these dogs" style threat, unless you're continuing to sustain an atmosphere of genuine inquiry and an air of being receptive to learning or changing your own views (it really helps with that if you genuinely are looking for answers and are open to the possibility that along the way you might revise your own position at least a little).
3 - Help Her Find Solutions
Once you've got her acknowledging that the dog behavior is a problem, ask her about what she thinks she could do to help solve it.
Your solution is "don't feed the dogs", but your actual problem isn't the dog-feeding, it's the dogs being in the neighborhood and becoming a possible threat.
She might decide that she wants to proactively train the dogs to only take food from her and not expect it from others (for example, by never giving them food except at a very specific spot in the apartment, or outside the apartment complex, while gradually teaching them to sit on her command in order to receive the food) or something else.
Either her solution solves the problem, or it doesn't. If it's obvious the solution doesn't solve the problem, apply this same process to examining possible flaws in the solution and other alternatives.
Or she'll decide on her own to stop feeding the dogs.
4 - Wait and Repeat
There's a very good chance that a single conversation will not significantly change things. You'll need to be gentle but persistent, keep building the actual connection and occasionally bringing up the issues.
Very often, the seeds you plant will take time to grow. Also, sometimes people exhibit gradually greater resistance to an idea in the lead-up to switching over to accepting it. Why this happens is outside the scope of this answer, but you need to understand that first and foremost, you need patience. Socratic Method -induced cognitive-dissonance-like resistance looks like the opposite of progress but it's just the person processing things.
But if you get impatient, if you start regressing into proclamations and assertions of her being wrong or into insistence or demands for a specific solution, you will antagonize her against those ideas, and disagreement-galvanized resistance is the opposite of progress.
Summary of General Principles
- Look to genuinely understand the person's position. If you still can't relate to it, you're not done.
- Socratic method questioning actually gets people engaged in proving themselves wrong, while disagreement gets them engaged in proving themselves right. And you'll do better at it if you are genuinely trying to understand the side you disagree with and open to finding possible errors in your side.
- People resent obvious demands or limitations being imposed on them by others, but once you've helped them see a problem, they are often able to come up with their own solutions that might be just as good or better.
- Mind changes take time, are often turbulent.
You need to understand what you're asking to do - you're looking to get a person to change their behavior, which very often (and I think this is a case of that) requires changing deep cognitive habits and beliefs. You can open a door for the person to do this, but that level of self-review and change is uncomfortable for many people, and it is your responsibility to provide them with a few of the other side of that door - if you're correct and there's less obvious errors on your side of that door, and what you show them suggests that there's a way for them to be happy and content with life and themselves if they move their perspective there, only then there's a good chance they'll walk through to your side.
The onus is on you to really understand what the person needs from their cognition and beliefs and provide usable alternatives using the techniques above accordingly.