1x3 would be fine for cross braces, that table would likely be able to take your weight, mine is constructed similarly (ladder rail construction, 1x4 for frame and cross members, braces I think I used 1x3), and I have gone on the table and I am not a lightweight, my legs are 2x2.
As far as warping goes,there are a couple of pieces of advice I have.1)No matter where you shop, don't get pine, get a hardwood of some sort. It will cost more, but they don't warp as much IME. Kiln dried pine should be fine, but these days honestly the quality just isn't there, especially at a big box store (Lowes and HD to me are neck and neck in terms of race to the bottom with that). I would recommend a lumber yard, I got my lumber there and while it was more expensive than the big box store, it was good quality and has held up. My basement has temp swings, and does have humidity swings even with a dehumidifier, and the word has held up beautifully. I used a 1/2 plywood top as well.
