Fielders should wear a glove on each hand or a glove that can be switched on the run.

It's much more natural and instinctive to catch something with an open hand on the same side as the object.  Using the same hand all the time even when forced to backhand is unique to baseball and unnecessary.

http://mlb.mlb.com/mlb/downloads/y2014/official_baseball_rules.pdf

1.12 catcher ... leather mitt not more than thirty-eight inches in circumference, nor more than fifteen and one-half inches from top to bottom ...

1.13 The first baseman may wear a leather glove or mitt not more than twelve inches
long from top to bottom and not more than eight inches wide across the palm, measured
from the base of the thumb crotch to the outer edge of the mitt...

1.14 Each fielder, other than the first baseman or catcher, may use or wear a leather
glove.
_______________________________________

I don't see anything, other than the usual lack of imagination, that prohibits or precludes a fielder from wearing a glove on each hand.  While the word glove is singular, that could easily be interpreted as a single glove per hand.

Outfielders would probably benefit most but infielders might also.  Particular circumstances, such as two outs, enhance the benefit.

The usual reactions to this suggestion include:

1. That's physically impossible.

2. What if he has to throw?

3. I'd rather backhand the ball.

Argh.

Other than catching a baseball, the normal way of catching is to use the hand closest to the object in the open position.  What football player catches the ball backhanded?

If the fielder needs to throw :

a. if the ball is to his non-throwing hand, simply drop the glove on the throwing hand and proceed as usual.

b. drop the glove on the on the non-throwing hand, catch the ball, remove the ball from the glove with the non-throwing hand, drop the glove, place the ball in the throwing hand, throw.

Only baseball people would need an explanation.  Thinking is so constipated that dealing with any change, no matter how logical, is excruciating.

Jim Abbott congratulates a player
from the Challenger Phillies
from Middletown, DE
on the South Lawn at the White House
July 11, 2004 by Paul Morse
via Wikimedia Commons

An alternative: a glove that can be switched on the run.  Some of the few switch pitchers have tried such a glove.

What!?  Switch while running?  Impossible!  Not if the fielder, you know, practices.

Find some moving images of former pitcher Jim Abbott.  Abbott played parts of ten seasons from 1989 through 1999 (not 1997).  In 1991 he won 18, lost 11.  He pitched a no-hitter.  He threw left handed because he does not have a right hand.  He placed his glove on the stump of his right arm, threw and put the glove on his left hand in time to field.  He then reversed and threw out runners.

For fielders with two hands either of my suggestions should be pretty easy in comparison.