You can disable linear interpolation on the texture (use GL_NEAREST). This gives a pixelated look though. A common way to deal with transparent images:

1.) Use a background color that is nearly the same as the average sprite texture. Your green space marine could have a greenish background as well. If you use black, pink or another different color, you risk the edges getting blended with the sprite-background as well if you use (linear)texture interpolation. And that is probably what you want to prevent a pixelated look. Interpolation means that each pixel also grabs somewhat color from it's neighbors (thus also the background at the edges) to smoothen the look. Older games like Doom didn't do this, which is why you didn't get the edges either.

2.) Use an alpha channel to mask the background:
- Copy your sprite to another (grayscale) texture
- Use a very different background color this time. In case the sprite is pretty bright, use a black background. The trick is to make a 2 colored black/white image first. You can use threshhold / stretch filters for this.
- Now that you have a perfect mask, you can blend the edges inwards. With a small gaussian blur for example. But be careful not to generate grayish pixels outside the mask. What you could do BEFORE blurring is removing 1 pixel at the edges. Maybe with a "narrow" / "shrink" filter or something.
- Now the edge pixels are somewhat grayish, meaning that they fade out.
- Depending on how your painting program works, import the grayscaled alpha texture into your original sprite.
- Save as a 32 bit TGA, PNG or DDS (make sure it supports alpha channels)
- Do NOT use compression! Compression messes the edges up again due quality loss.

3.) Import the texture. Make sure the import and output formats are RGBA, not RGB! You'll need to load and store that alpha channel as well. Enable interpolation on the texture (GL_LINEAR, mipmap, or something else).

4.) When rendering, enable blending.

That should do the job
Rick