How do you access the database?

Answer

1.  This is perhaps the smallest issue here for a simple reason: most good Drupal developers shouldn't actually touch the database on a site. Drupal is so powerful that you can build a very powerful system without ever touching the database. Nonetheless, the database tool of choice will help you gauge their technical astuteness in general. Also, a lack of technical astuteness in this area in particular begs the question: "What happens if there's a problem with my database-driven site then?"
2. We use SSH/command line to access the database. This is best, as with the previous question; this demonstrates a comfort with a more powerful tool. This basically rules out the possibility that your developers are database novices. Even if your developers don't prefer this as tool, you might ask them how familiar they are with it; familiarity with it is a great sign.
3. We use a desktop based client that connects through SSH. This is also good. A good tool here typically packs similar power (sometimes more) to what you can expect from a command line tool. However, unlike the SSH option above, this doesn't completely rule out the possibility of novice developers; that they connect through SSH probably means they're concerned with security, which is a good sign.
4. We use a desktop based client that connects directly to the MySQL server. This is ok. It's not quite as good as the above option, especially from a security and flexibility standpoint. Connecting directly to the MySQL server probably means they're not aware of the security implications of that at the server level, which could be a red flag. However, it can be just as powerful.