1961 was one of the final years that US silver quarters were minted. These coins aren't rare, but they do have value as silver bullion.
Type: Washington Quarter
Face Value: $0.25
Composition: 90% silver, 10% copper
Silver Weight: 5.62 grams
Total Weight: 6.25 grams
Current Silver Bullion Value: $3.22
The US minted the 1961 quarter with no mint mark and also the 1961 D quarter. The mint mark can be found on the reverse side of the coin.
The value in circulated condition is worth at least its weight in silver. The scrap silver value for this coin is $3.22 as of September 19, 2019. This scrap value is calculated from the current silver spot price of $17.83 per ounce.
Over 37 million 1961 quarters with no mint mark were issued so they aren't rare. These quarters essentially only carry a premium in uncirculated condition where each coin is valued at around a $4 premium over the scrap silver value.
The 1961 D quarters are even less rare. Only in uncirculated condition do they command a premium of around $3 over the scrap silver value.
Proof coins are available and they are valued at around $10 each. Graded proof coins with a grade of PR 67+ will sell for more. There were about three million collectible proof coins minted so they aren't rare.