Should I keep my bitcoin-qt wallet running all the time? It can take a day or more to get in synch, so it's somewhat problematic to only start it when I want to transact.  But are there any security issues in keeping it running constantly?

What's the general practice for those using local wallets?
The first time you load the client, it can take some time to download the whole blockchain (currently something like 15 GB). However, if you close the client and then reopen it, the client will only download the new blocks from the blockchain that it hasn't seen yet. The previous part of the blockchain is already stored on your computer.

To catch up a day's worth of blocks should only take a minute or two. It's perfectly reasonable to close the client when you're not using it.