Nevadagnome,
First, WELCOME to the fracus!!
Second, the "master cylinder" is on the handlebars (front brake) and the "slave cylinders" are in the calipers on the brake disc (wheel). Assuming you mean the leak is somewhere around the caliper slave (front-wheel right-side caliper), there are maybe three prime sources of a leak to look at.
The ST1300 has a very unusual and complicated "linked" braking system where the braking force on the rear caliper, and on the left front caliper, will apply hydraulic pressure to the other, front or rear braking system. That's why there are three brake lines to the front left caliper and rear caliper. But your front right caliper is leaking which makes this easier.
1. You may have a leak in the flexible "rubber" brake line above the caliper that is running down the line to the caliper. These older "black rubber" lines are a well known problem. DOT4 brake fluid attacks paint and plastics (wipe off all spills
immediately with a wet rag and wash the area with water if you don't want permanent damage) including the "rubber" brake lines. At about 15 years old they begin to fail. Sometimes it weeps, sometimes it swells the line, sometimes it clogs the ports and passages in the caliper with decomposing bits of "rubber" from the inside of the brake line. The only solution is to replace ALL the flexible brake lines on the bike with Galfer or Spiegler PTFE lines (braided SS). You will need to do this at some point anyway.
2. You might have a leaking slave cyl seal. It is usually the aluminum cyl wall that corrodes and lets the seal leak. Replace the caliper.
3. You might have a leaking fitting, a banjo seal, or the bleed nipple not seating. If the brakes have been serviced recently this might be the problem.
To find out exactly where the leak is, wash the bike thoroughly, especially around the front caliper, then without using either brake, let the bike dry thoroughly. Then apply the front brake sitting still, starting with light pressure, and looking very carefully for any signs of the leak. If the leak is very small it may take a few minutes of pressure to see the leak. Verify it by washing that area again and blow-drying the area thoroughly, then re-doing the pressure test.
Let us know what you find.
You'll want a Haynes manual for the ST (eBay) even if all you do is drive it. Lots of info there. Clymer is better but they don't have one for the 1300.
And post some pics. No pics means it didn't happen.
if needed.
