My TH350 in my 1960 Chevrolet pickup leaks fluid from where the speedometer cable screws into it.
Anyone have any tips or suggestions on how to fix this problem?

if i recall there is a seal that pushes in on the cable side of the joint.it like a big rubber bushing

Just saying. Take care. Tom Worthington

The shaft of the gear is often worn a lot and allowing more fluid out to the seal.

If I remember correctly, there is room to double up the lip seal. Gear shaft wear can be a problem, as mentioned.
Vehicles that sit idle often have drips from converter drain back over filling the pan, putting pressure on seals not meant to be submersed.

I have a handful of those seals in my toolbox, but they're probably all over 45 years old.

Remove the speedo gear "bullet", if it's steel it's probably OK to just reseal. If it's aluminum replace it and the gear.