first, boers have many nipples; some work and some dont, and sometimes more then one works on one side and sometimes the kids cant latch properly because of the other teats. i had one that her kids used to suckle from two nipples at once because one was practically on top of the other so...
as the one person said: warm up the area, i actually used olive oil, do a gentle massage (like u would do on your breast, so as not to bruise the tissue) all teh way around, and then try clean and dry your hands, and then try gently to strip out some milk; check for a plug, make sure she is relaxed, give her some munchies while u mess with her, and also try to let one of the kids suckle from that side. check for discharge, or purple bruising (thats an emergency if u dont deal with it)... why would u have to put her down? mastitis is treatable. jsut make sure u milk her after u touched all healthy animals AND the healthy side. a bad breast is the last thing to do before u wash and clean your hands (fingernails, everything, with good soap.)...
and if that is the teat that we see, i've had those before... the teat doesnt work well, and the kid cant latch well... u will have to milk her out a bit and make sure the at least one kid is eating from that side. sometimes kids will only suckle from one side and the other side jsut stops 'working', just like with us, there is always one side that the baby prefers for whatever reason, so once a bit full, the baby wont drink from the 'difficult' side, so also for goats. OR u can milk out that side and use it or save it or feed it to the dogs or whatever...