Legal Question in Constitutional Law in Missouri

The language you cite says only that the states can't create any forms of currency besides gold and silver coin. It doesn't say that they can't *use* other forms of currency, or even that they have to use currency at all. Payment by check is perfectly legal.

Note that this clause only limits what states can do. It doesn't apply to the federal government, which is authorized to create paper money and to make coins out of other materials. States are free to use the currency created by the federal government. That, of course, is what they all do, since no state makes its own coins.