Even though the alternator is capable of 37 amps, it won't be running at that rate rarely, if ever.
How much of a load do you currently have? With the car motor turned off, turn on everything electric. Lights, horn, turn signals, wipers.... What is the maximum reading when using everything. If it doesn't go over 20 amps with no charging going on, it also won't with a higher capacity alternator while charging.
On the charging side of things, I have a Nu-Rex high output alternator and a 20amp meter. Right after starting, the charging grate will momentarily swing up a little past 10 amps and then settle back down to near zero while running. If your battery was really shot, you might get over 20 amps on full charging, but still, probably not.
For other things, such as shorts, etc, hopefully you are running with a fuse, and if your fuse is 30 amps, it should not let your current go over 30 amps anyway. 30 amps should not kill a 20 amp ammeter. They are more rugged than that.
My opinion is that you should probably never see any current issues. An alternator with a capacity of 37 amps is its capacity, and does not imply that it will always be putting out its maximum current.