I agree with pretty much everything listed in all of the posts, so I will list some things in particular I would look for.
First, I want somebody who is at least minimally competent in the game.
How well do they know the game? Do they know mechanics, strategy and the rulebook? If they don't know mechanics very well (which includes me for pitching), do they know they don't know and therefore won't do damage? Are they looking to educate themselves in the game or do they assume they know all they need to know from their own playing days? They need to understand strategy well enough that they aren't making decisions that are wrong 100% of the time (sac bunting down 3 runs in the 7th inning, for example, or failing to throw through to second on a 1st-3rd in the 7th inning with a two-run lead). Any decent coach will have read the NFHS or ASA or other organization rulebook at the start of each season. The number of mistakes made by coaches (and even umpires) because they refuse to read the rulebook is astonishing.
Second, are they passionate?
I want a coach who loves the game and wants to coach. I don't want someone who views him/herself as just doing everyone a favor and who is taking one for the kids. Those are certainly good people with good intentions, but too often their lack of passion for what they are doing means that they will be a babysitter and not a coach. If I'm a parent of a serious softball player, I would much rather risk having a coach who may appear to be in it for himself but who is passionate, rather than the coach who says he is doing it just for the kids, but who does nothing more than just show up unprepared for practices and games.
Third, are they not insane?
Having done this for awhile now, I strongly believe we have to give coaches some slack. We make thousands of decisions every day we coach, in that every time we open our mouths, we risk saying the wrong thing or something that someone thinks is wrong or insensitive. The act of making out a lineup is guaranteed to bring detractors. If you have 12 girls on a team, what are the odds that any one person is going to agree with the exact lineup you put on the field, including the batting order? Because coaches are under such a magnifying glass, they are often held to a much higher standard than people in other jobs or positions. As it pertains to volunteer or even school coaches, it's just not realistic to hold them to a standard that's expected even for people in their full-time jobs. As a result, I just want to know that they aren't crazy or strange enough to where their behavior is not acceptable. Of course, that would include any mistreatment of players and any repeated embarrassing episodes. Also, they generally need to be solid role models. No drunks, drug users, etc. In my book, a good role model as a coach is one who is organized, punctual, prepared, hard-working, able to set aside which players he likes best when it comes to how the girls are treated, sets an example of how to treat people, and anything else in that whole vein. I'm not big on the coaches who constantly talk about "the game of life" and "life's lessons" and all of that. Don't get me wrong, of course we want to teach those things, but I'd rather have coaches who just set the example instead of constantly talking about it and who then come totally unprepared to practice and never bother to read a rulebook.
As Sammy said, the list could go on and on.