There's slightly more to this picture than meets the eye, the texture coordinates are not the only factor in where the texture gets sampled. In your case I believe the blue is probably what want to have.

What you ultimately want is to sample each texel in center. You don't want to be taking samples on the boundary between two texels, because that either combines them with linear sampling, or arbitrarily chooses one or the other with nearest, depending on which way the floating point calculations round.

Having said that, you might think that you don't want to have your texcoords at (0,0), (1,1) and the other corners, because those are on the texel boundary. However an important thing to note is that opengl samples textures in the center of a fragment.

For a super simple example, consider a 2 by 2 pixel monitor, with a 2 by 2 pixel texture.

If you draw a quad from (0,0) to (2,2), this will cover 4 pixels. If you texture map this quad, it will need to take 4 samples from the texture.

If your texture coordinates go from 0 to 1, then opengl will interpolate this and sample from the center of each pixel, with the lower left texcoord starting at the bottom left corner of the bottom left pixel. This will ultimately generate texcoord pairs of (0.25, 0.25), (0.75,0.75), (0.25, 0.75), and (0.75, 0.25). Which puts the samples right in the middle of each texel, which is what you want.

If you offset your texcoords by a half pixel as in the red example, then it will interpolate incorrectly, and you'll end up sampling the texture off center of the texels.

So long story short, you want to make sure that your pixels line up correctly with your texels (don't draw sprites at non-integer pixel locations), and don't scale sprites by arbitrary amounts.

If the blue square is giving you bad results, can you give an example image, or describe how you're drawing it?

Picture says 1000 words: