A “filing basis” is a trademark registration term. In the United States, a trademark registration application must be filed on one of several bases. It is not enough to think of a trademark that you like and then apply to register it – you must satisfy at least one filing basis when applying. You can think of this as one of the minimum requirements for a successful trademark application.
The bases are: actual use in commerce, an intent to use in commerce, a foreign application for registration in a treaty nation, a foreign registration in a treaty nation, or registration under the Madrid Protocol. For U.S. trademark owners, the most relevant bases are the first two: actual use in commerce or an intent to use in commerce. The other bases are more relevant for non-U.S. trademark owners.