There are a few different questions here. I will try and address them.
Yes you need the 32-bit client dlls to talk to Oracle as MapInfo Professional is currently a 32-bit application. All 32 bit applications need this. The client does not have to be the big formal client, the "instant client" (Oracle's name) will work fine. The end result that matters is two fold:
Can the software (easyloader or Professional) load the Oracle client dlls? (Must be 32-bit and they are on the system path)
Can the software use alias names for the database connections? Generally this means that the tnsnames.ora file can be found by the oracle software so that it knows what a simple alias like "Oracel11" that you might enter in the connect dialog actually evaluates to. This is not a requirement as one can specify the longer form but it quite common.
So if Professional is connecting is it via OCI or ODBC? The ODBC connection can only be used for non-spatial. You just use the OCI connection type. That should be available when and only when Professional can find the dlls as specified above. If this is not the case, check your system path and see if the 32-bit location for oracle was added. If you are using OCI from within Professional, I am stumped. I would call support.
For Easyloader you say the Oracle connection is grayed out so that is why I am thinking something like above is amiss.
Eric Blasenheim
Pitney Bowes Software