The tiny hole at the bottom is there by design, to let hydraulic fluid out if a leak develops in the slave cylinder for the clutch system.
Because you have stated that you see fluid leaking out there, it's about 99% probable that you have had a failure of the clutch slave cylinder. That is a part that you can replace yourself, provided you are a contortionist.
I had to replace that part last year on my ST 1100 (it's much easier to replace on a ST 1100 than a 1300) - here's a link to a post that will give you an idea of the
concepts (and a few of the pitfalls) involved in replacement and then re-filling and bleeding the system:
ST1100 - Replacing the Clutch Slave Cylinder.
Regrettably, the task is more difficult to carry out on a ST 1300 because the slave cylinder is more difficult to access - it's not at the front of the engine. Some forum members have successfully carried out a replacement. Here is an article by
@MaxPete about ST 1300 clutch slave cylinder replacement:
ST1300 - Clutch Slave Cylinder Replacement. That article and the 7 pages of discussion and additional information that follow is really scholarly - by the time you read it all, you'll know everything you need to know to do the job.
My guess is that the parts cost will run you somewhere between $100 and $200 - but that is just a guess, based on my recent experience with my ST 1100.
Michael