I have mostly hostas or daylilies with my bulbs. For other areas I just fill in with violas/pansies and then allysum/snapdragons/other little annuals. They don't really cover so much as distract.

I also use daylilies. Since you want a border you could use Daylily Stella D'ora. It blooms most of the summer and it is short and would cover the leaves of your bulbs.

I kind of like the daylily idea, the stella dora is beautiful. There are quite expensive though if I was to plant them along the whole border. I bought one daylily recently (not sure what kind) and it was $7 just for one root :(
I am also thinking allysum, I can just plant the seeds.

Daylilies multiply rapidly. You divide them to get new plants. You could buy just a few Stella and gradually fill in your border.