Each is OK, but the meaning and emphasis are slightly different. And the specific emotional content is not necessarily clear in the first two.
I can't see you crying.
This is a little awkward. It seems to be leaving something out. It might be used when a person is trying to say that they are unable to endure the experience. Or, possibly, that they are unable to remain passive during the experience. An example might be "I can't see you cry without doing something to make you feel better."
Alternatively, it might be indicating the speaker is very callous and hard. They are stating that they literally don't see the other person crying. They can't see it, so it does not affect them. A version of this is "I'm neither sugar nor salt, so tears won't work on me."
I can't stand to see you crying.
This is little more clear. The speaker is saying that they cannot stand, meaning they cannot endure being idle, when they see the other person crying. They are claiming directly that they will try to do something about it.
Note that it does not express any specific emotion about it. The speaker might try to comfort the crying person. Or the speaker might turn away so the sight of the crying person is removed.
I don't like seeing you cry.
This is specifically indicating the emotion the speaker feels due to the sight. But now, the action has been removed. Maybe the speaker is able to stand idly and endure the sight, or maybe he will do something about it.
To choose the correct wording you need to figure out what action and emotion you want to express. It may be you need more than one sentence to get both the action and emotion.