We're growing asparagus and it comes up in small amounts at a time. How should I save it so that we get enough for a veggie side dish?

We grow and eat a lot of asparagus. It is fine for few days in the fridge, in a plastic bag. If you picked it 3 days ago, it is still probably much fresher than you'd get from the store.
Don't forget that asparagus is a very heavy feeder and especially likes lots of nitrogen.
As for harvesting, once the spears start to get small (smaller than a pencil in diameter) it is time to ease up on the picking and let the plants grow out. Leave the plants alone, keep feeding and don't cut the fronds until they are brown and dead, usually very late Fall.

I don't know when you planted your asparagus, but if you take too much from the small amount there is, the plants won't be able to manufacture enough food for the planting to increase. For a new bed, none should be taken the first year and only a small amount in the second. Even when the bed is established, some asparagus must be left. It does take awhile to build a bed.

I,too, planted 21 crowns only to find they were sparse providers,yet healthy plants. So, I have gone back and planted more,hoping the additional 24 I planted this year will produce a meal one day. Asparagus gardening( I won't even imply "farming" ) is truly a test of patience and good soil cultivation. My compost pile stays skimpy as I am constantly feeding the asparagus bed. Stay in there,FranA ; one day we'll have the last laugh and enjoy our harvest...ch

it takes awhile to establish an aspara
gus garden

I've seen asparagus sold at the farmers' market sitting in a jar of water like a bouquet of flowers would. I tried it at home and it seems to help the asparagus stay fresh for a few days.

I don't think you should. It starts to lose its flavor as soon as it's picked, and it's not a good keeper. Some people freeze it; some people say you can keep it longer if you keep it wrapped in damp paper towels. I think the joy of fresh asparagus is really fresh asparagus.