I think I may have a similar problem. Is this something that I can fix myself? I looked at the video posted and appears not to be a major under taking. I am a novice mechanic, pretty good at following instructions.

Doc, a proper Hanes manual will go a long way to helping you with your mini, from basic concept, to repair

All Mini engines use the transmission case as the oil sump for the engine - when you add oil to the engine, it drains right into the gearbox. Therefore, the engine oil dipstick tells you how much oil is in the gearbox. Keep the level between the two marks on the dipstick. Use 20W50 oil, a heavier oil that protects the gears as well as the engine. If you change your own oil, there is one drain plug at the end of the transmission case. When replacing this plug tighten it carefully. The case is aluminum and the threads of the drain hole are easily stripped.

How do I check and fill the tranny fluid?
Thanks guys!   HUGE HELP!

you'll know when you pull out the old seal:  if there is aluminum behind it that will make it quite clear that the spacer which came with the kit will not be required

Aww man..... thanks so much.
   I was worried sick, I thought it was a LOT more complicated to change that seal out.
So, how do I tell what engine I have?  The newer one with the extra blocker seal or the older that needs the whole kit.
 Thanks! 

Very common leak on rod change transmissions.

There's a kit for it
//www.minimania.com/part/MSSK050/Mini--Mini-Cooper-Rod-Change-Anti-Oil-Leak--Seal-Kit\

here's how to disconnect the shift linkage (don't worry about axle part-- leave them there.)
https://www.youtube.com/watch?v=OBgKFtLxk00


This video is the anti-leak kit where the guy is actually pulling out an old anti-leak kit and replacing it with another kit.
https://www.youtube.com/watch?v=61FMWca5AM8


But if you have a later DAM5626 casting, you don't need the whole kit-- you just need the seal-- because the transmission has an extra collar built in that the alloy part in the kit compensates for in the older boxes.

//www.minimania.com/part/AHU1672/Austin-Mini-Cooper-Rod-Change-Shift-Seal

So I went to get my '70 out of the garage and its got a puddle under it.    (1275cc)
I think I have it narrowed down to the linkage seal where the shifter rod goes into the transmission.
 The seal was cocked a little and dripping.  The collar looks like its slid back.
First can somebody comment if this is common or expected?

Should this seal have a little spring on it?
 Can somebody post a pic of their linkage  at the tranny end (in neutral) and I can see the position of that connector collar?
Kind thanks.

Any help is appreciated.