A penny weighs 2.5 grams. 37 pennies weigh 92.5 grams or 3.26 ounces.
For first class letter mail, the USPS charges 37 cents for the first ounce and 23 cents for each subsequent ounce (or portion of an ounce) so you're looking at $1.06 in postage for your 37 pennies.
Thanks to
The U.S. Mint and
The USPS