Is the water dripping from the gearbox end? If so, it likely is the thermostat housing. It is a common problem. New unit is £140 from Mini in the UK. The forum seems to agree you should change the attached crossover pipe while changing the thermostat, it's relatively cheap and it's a common failure too.
I've done 2 on my own cars this year, I think a garage could do it in a morning so $1400 does sound steep to me.

 

@North021884 Thermo housing is a common issue and by now I would say you are due a failure! But $1400 to replace, far too much, $400 plus parts for the housing and circa $600 if you include the cross over pipe.

 

I hade thermostat, water pump and crossover pipe done this year for around $900 , at a BMW/Mini specialist.

 

As above, yes it is common.

I did ours with the cross over pipe at the back in 1 hour, plus 30 mins to bleed it. As advice, only use a genuine BMW rear cross over pipe. Thermo housing I went for genuine Febi - be very wary of the copy crap out there. I paid £65 for the pipe from BMW and £125 for the Febi housing - so say $250 for parts and max 2 hours labour

 

Same issue here on a 2012 JCW. Quoted $1800 mostly labor because of the time to dis and re assemble to allow access for the housing unit.

 

I wouldn't normally comment on garage pricing since they need to be able to work out their own prices, but think $1400 is too much. Maybe they’re overestimating how long it’ll take and maybe they’ve never done it before (not meaning to say they wouldn’t do a good job). I’m completely new to working on R56 engines and I did the thermostat housing for the first time a week or two ago and despite never having done it I only spent a few hours which included getting everything set up (I don’t keep my tools near my car and didn’t know for sure which tools I’d need, and I had to setup mosquito repellants too ).

It’s a tight repair but not difficult. I recently did the oil filter housing on a 2006 Audi A3 2.0T, and compared that the Mini is spacious and easy to access!

 

Hi, I too am new to Minis but I am fast learning about common problems, and this comes up again and again.
As Bigphil555 says above not a hard job, so I would say $1400 is far too much, I could fly over from UK and do it for that price, only kidding.
I would take it to a mini specialist at least they will have a better idea of what is involved.
Good luck.