Some of mine are dusty [am not suggesting that is what is on yours ] but do you use things like talc or dry shampoo in the bathroom?

If it is clumps it could be woolly aphid.

Give it a warm shower and leave it to drain and see if the powder has washed off.

Need a clearer picture really.A close up of one of the white bits in a better light would help with identifying it. Try Seaburngirl's suggestion first. If it doesn't help proceed as follows: if you can lay your hands on a small brush, eg a child's paint brush, dip it lightly in methylated spirit and dab one of them .If it disintegrates and a little bit of red appears it will be mealy bug and you can treat it all in the same way. If the plant continue to look healthy that's good. If it looks unhappy - and this may not happen until next summer it may have spread to the root system, in which case get a pair of thick gloves and tip it gently out of its pot so you can see the roots. If the roots show the same stuff the whole root system (and the inside of the pot) will need treating with meths, and then it will need repotting in new compost.You can buy small bags of potting compost specially designed for cacti.

Looks like mealybug and it is very hard to control. If you have other plants near this one, get them to another location and check them periodically for infestation. Wipe the whole windowsill with a 10 part water/ 1 part bleach. If it were me, and the plant had some sentimental value I would follow Steragram’s advice. Otherwise, into the garbage it would go.

It would be a pity to sling it though because its not a youngster any more.Just had another thought. Out with those thick gloves again and after the meths treatment detach one of the offshoots, preferably one with no sign of the problem.Pot it up in dry cactus compost and hope for the best. Wrong time of year really but you might be lucky. Don't water it though before it's grown any roots or it will rot.