I also agree that sticking them in water helps - snip the ends as though they're cut flowers and put them in an inch or so of water.
I also sometimes chop them and freeze them in ice cubes - then just throw an ice cube into your sauce or whatever and you've got fresh herbs in it!
If you have any outdoor space at all it's really easy to grow herbs in pots. Rosemary is really hardy and will last all year (mine has survived everything from snow to heat waves). Basil is OK but I've had trouble with slugs eating it. At the moment I've got: basil (purple and normal), rosemary, sage, tarragon, thyme (also very hard to kill), parsley (eaten almost entirely by bugs or critters - not sure which), mint, chives (quite hardy), and peppermint all growing in smallish pots just outside the kitchen door. The rain hasn't been great for them, but otherwise they're pretty easy to deal with).