Yes, you can write to a sde database where the sde service is not running by using GEODATABASE_SDE writer and pointing to a sde connection file. The legacy SDE30 writer, on the other hand will not be able to write to your enterprise geodatabase if your sde services are not running.

I believe the question is a bit off:

The database needs to be up and running, the SDE services that connect ArcGIS to the database do not need to be running. The connection can be made by FME.