Firstly, the pixel density will depend not only on the resolution of your texture and the screen, but also the field of view. A narrow field of view will result in less of the skybox filling the screen, and thus will zoom into the texture more, requiring higher resolution. You don't say exactly what FOV you're using, but I'm a little surprised a 1k texture is particularly blurry, so maybe it's a bit on the narrow side?
Oh, and before I forget - you should be using compressed textures... 2k textures shouldn't be that scary.
However, aside from changing the resolution, which obviously does start to burn through memory fairly quickly, I generally always combine the skybox with some simple distant objects.
For example, in a space scene I would probably render a fairly simple skybox which only contained things like nebula, etc., where resolution wasn't critical. I'd perhaps render at least some of the stars as sprites, where the texture density can be locally higher. A planet could be textured geometry.
If I was rendering a more traditional outdoor scene, I could render sky and clouds on the skybox, but a distant horizon as geometry. A moon in the sky might be an overlay.
There is no one standard answer - a variety of techniques can be employed, depending on the situation.