Photo captions, and titles in general, are not normally expected to be complete sentences. Full grammar rules do not apply.
In general, I'd say you only need a "the" at the start of a photo caption if you are trying to clarify or emphasize that this is a particular instance of the thing or the only one of the thing. But the difference would be subtle.
In the examples you give, captioning it "The River" would imply that it is some river that you have just been talking about, or that this is the greatest or most important river in the world. Like an Egyptian might label a picture of the Nile, "The River", but an American would probably not, at least not without some explanation. If I was putting a caption on a photo that was just some random, "typical" river, like from a collection of pictures of generic geographical features, I would likely label it just "River". If it mattered what river it was, like I was labeling pictures of places where events that I am describing occurred, I would give additional information, like "Mississippi River" or "River Behind My House" or whatever.
Likewise, a caption of "Sitting and Standing Women" would be appropriate on a picture which is intended to be artistic. Umm, "The Sitting and Standing Women" would be unlikely, but possible if you are contrasting it with some other group. Like picture number 1, "The Sitting and Standing Women"; picture number 2, "The Reclining Women"; picture number 3, "The Running Women".