Will it? The API is presumably already used by the front end. If the API is being consumed, it’s safe to say the front end will be consumed less, hence equalizing the support costs. Plus, rendering a front end requires calls to get data the user might not want at the moment, whereas an API can provide fine tuned access to only the needed data, hence potentially saving money.
A public API is going to incur additional testing, documentation, and support. Charge accordingly.