
When I want to check continuity across a connector, I will touch one of the meter probe wires to a bare spot of the wire where it first enters the connector. The other meter probe wire is then simply touched to the bare end of the connector. The meter readout will tell you if there exists any 'resistance' across the two points. If there is, then you have some corrosion inside the connection. If there is no resistance, the connection is a good one.
If I can't get onto a bare spot of the wire where it enters a connector, I will cheat by using a second tool I have that contains a very sharp metal end. That tool is my 12V test light, made to puncture through a wires' insulation. Once I've inserted the sharp end of the 12V test light into the wire, I clamp onto the sharp end with a small wire I've put together which has little alligator clamps at both ends. The other end of that wire, I will clamp to one of my meter probes. etc...