Baseball helmets for infielders?

Absolutely not. This is a silly rule to implement. 2 very good reasons. One is rate of injury, two is cost. This goes back to the everyone plays equal rule where coaches are forced to play kid in the infield who have no business being there. When you decide who should play where, safety is a factor. If you have a kid who does not pay attention or cannot catch very well, you don't want to put them in the infield where they are more prone to sharply hit balls.
Second is simply cost. You madate this and you've added at least $100 to the already strapped budgets of high school sports teams.
At some point you have to say that there is going to be some inherant risk of playing sports. You can't put a suit of armour around each kid for fear they may get hurt. It's unfortunate that a kid gets hit in the face or head, and yes it can happen, but not at a high enough rate to mandate equipment.

Yes, especially with young players in smaller community leagues. Many struggle catching, and are scared of the ball, yet because coaches are forced to play kids equal, they put these kids at risk. The ones who are real difficult to deal with, are the parents who think their kid is pretty good, but they really aren't.

One thing I think coaches and parents don't see is the value of being a good outfielder, even at the little league level. For some reason there is this perception that the weaker players are always put in the outfield. In reality, when kids get to be 13-15 years old, many balls are hit in the outfield and the successful teams have kids that can play those outfield positions well.

baseball helmets on infielders would also teach our kids that they should be very afraid of a ball coming their way. The best way to handle a line drive or bad hop is not to let fear enter the mind, rather, to become comfortable and aggressive with this key part of the game. Charging hard hit balls is an example of this. What would a kid do that has relied on helmets or face guards while playing infield in his youth when he gets to college or pro's, they might feel naked without that protection, further endangering them on the field when they close their eyes or duck for cover. While something needs to be done about preventing these injuries, I don't think armoring the kids is going to be the best answer.

I would really like to see some statistics detailing the number of softball injuries to the head and face by hit balls. Not bad hop grounders, but balls directly off the bats that hit a player. Many of these safety ideas come from an over protective mind that is trying to keep everyone from ever getting injured. Think about it. Have you ever seen this happend? Then think about the hundreds or thousands of innings you've watched over the years? Is there a real need for this?
Remember, one time long ago, football players wore leather helmets and very little padding. They tackled differently and it was more of a skill than just throwing your body into your opponent.

Larry, no amount of space on this blog can I respond to the feelings I have for your son getting hit in the eye. I'm sure it's been an ordeal that you cannot tell us enough about.
However, people die in car crashes everyday, yet we don't ban the use of the automobile? My point is there in a threshold of the percentage of injury that has to warrent the use of safey divices. My experience on this blog is that some of our writers of this site would like to see a day when no kid ever gets injured playing sports. An absolutely unrealistic goal.
You design youth sports programs and you try to be as safe and sound as possible within budgetary constraints. You can't equip each kid with a suit of armour to protect them from anything that could happen to them.

No one likes to say this but it also becomes a cost.   There is a risk/ reward in anything to do with safety.  A long, long time ago, football players and batters in baseball games did not wear helmets.  It became apparent that there was a high number of head injuries.  Football players were mandated to wear helmets, as were hitters in baseball.   Costs of football equipment skyrocketed, but there was enough need for the equipment to justify the costs. 

Now, moving it to baseball/ softball, there isn't enough data to suggest that kids are in  a high degree of danger without helmets in the infield.  Yes, some do get hurt unfortunately, but not at a high enough percentage to mandate helmets on all kids.