Depending on what classifies the transactions as fraudulent will determine next steps.

If the fraudulent transactions were due to the cardholder having their card stolen and then being used to make fraudulent purchases you should work with your merchant services provider. Within your gateway or shopping cart you have settings that allow you to request AVS, CVV, CID, Address Match, etc. and you can determine the velocity settings and how strict you want to be for each transaction. Some gateways also have country blockers to eliminate transactions that are coming from known organized crime or hackers specific to fraudulent transactions.

You should always report any suspected fraud to your merchant services provider, gateway or shopping cart provider along with the authorities in your local jurisdiction.

Setting your filters and tolerances really low will allow transactions to go through and generate revenue but when you do not validate AVS, CVV, CID or address match, etc. then you risk the chance of fraudulent transactions.

If you collect all the above information and it still ends up being a fraudulent transaction then it should be the responsibility of the card issuer and not the merchant or merchant services provider.