You should be able to setup the online database to only accept a connection from your IP address, which would limit the security issue somewhat. It is password protected too, so it should not present a major security headache - especially since the database is presumably only containing data that is available online anyway.
If you just want to read the data from the database, then you could do it via a web viewer in FileMaker as an alternative.
We have been running such a database linked up in this way for several years.
Good luck with it.