It needs to weigh enough that it counters the momentum/drag of the cable plus the net of whatever mass is going up. Keep in mind that cars going down add to that overall value while cars going up subtract. Also, the general opinion is for the station/anchor to be slightly above geosync so the net effect of the orbit on the station is to be pulling away from the earth (there is some wiggle room depending on how robust your earth anchor is and the mechanics of your tether with respect to tension vs. compression, but most models plan for a little net lift). In other words, you also attach to an anchor on the earth (which could just be a chunk of bedrock) to counteract that net force. Since the net force of the tether (not counting the earth tether) would be away from earth, any net loss of momentum would be regained from the earth’s spin (which happens whenever we launch a rocket right now). You could also have a spool at either end to maintain the desired tension on the tether while accounting for slight elevation changes due to net momentum loss or gain. On top of all that, the space anchor mass isn’t really dependent on the mass of the earth so much as it is on the net amount of mass being lifted or lowered to the earth and the amount of time you want to wait to return to it’s desired orbital altitude. And finally, if the tether was severed only the part whose center of gravity was below geostationary orbit would actually fall to earth - the rest would leave orbit.