When the baby goat is born, it may still be connected to the mother. If the umbilical cord has a pulse in it or is very full, leave it alone for a few minutes until the pulse stops and the cord collapses. There is fetal blood between the kid and the mother, and you don't want to give either a chance of bleeding too much should something go wrong. Don't worry, and don't hurry.
OR, you might want to tie the cord off near the kid (some say an inch or two from the kid's belly) and tie it off again a couple inches further down, and then cut or tear between the two tie-offs. Dental floss is good for this, or those plastic clips made for the purpose. Tearing with thumbnails or a blunt knife causes tears in tissue which clot faster so reduce blood loss, while a really sharp knife cut is more likely to bleed for a long time. Best to let mom do it or tie or clip first before cutting.
Mom usually eats the cord before I can get a chance to snip or dip. Never had a kid or lamb get sick from a dragging umbilicus so long as it was dipped in disinfectant early (I use iodine but have also used a moderately blue dilution of Nolvasan solution). You can dip whatever is there, or you can tie or clamp near the kid and tear with your thumbnails or cut, shorten, and then dip. Your choice.
The one time I did not catch a lamb being born, and came up an hour or two later and felt it was too late to dip the cord (Cord? What cord?) , the lamb started limping a day or so later. I described it to a more experienced person (animal husbandry professor, too) who suggested a shot of LA-200 to the lamb, and it stopped limping. My friends in Australia say that when they are lambing out hundreds or thousands of small ruminants in a day, they don't have time to dip umbilicuses, so they always have some sheep who develop "joint evil" or "joint ill" which is when bacteria get into the umbilicus from the soil and settle into the joints as an infection. So, a lot of sheep in Australia have arthritis. They live with it. We micro-manage and treat everything obsessively here in the USA, or have so few animals that we can grab them and prevent this condition, or cure it early, so we have happier goats and sheep!
Don't sweat it. If you see a lamb looking sore or limping, give it LA-200 or another long-acting oxytetracycline (spelling???) or whatever your Vet recommends and it will feel better. It is part of being a good herd-manager and having the luxury of time and the resources to do it.