They fight with sabres all the time. No musket. So why?
They were assigned to the Musketeer's unit. Unit names rarely designate the actual weapons - for example, there was a regiment of Fusiliers in the UK army in 1962, but they didn't use flintlocks (Fusilier is a word that means "flintlock shooter"), nor do the Grenadiers fight exclusively with grenades. And the Horse Guards... Or to choose another example, every modern cavalry unit uses transport other than horses.
Musketeers was an elite unit as mentioned by @lawson and the wikipedia page, and I believe that they are the equivalent of household guards for the king. The Musketeers were assigned to the King (Remember that the King actually had very few armed forces; most were managed by others, or by France, and separate from his command).
Also, Muskets are a pretty stupid weapon for urban fighting; although they ride all over the country, the unit is based in Paris, and designed for Paris.
While they fight with Sabers most of the time during the novel, they are members of a military unit called the "Musketeers of the Guard", which is where the name comes from. Notably all the musketeers in the novels are loosely derived from real people of the same name who were members of this organization.
Because the book shows very little warfare. In most incidents they were involved in, they conducted special missions, participated in casual scruffling or in the duels, where the muskets are useless.
Even in warfare the use of muskets of the time was very limited: they mostly were used for the first volley, and then the shooters switched to cold weapons, because the recharging the muskets took too much time.
Technically not a sabre, but a rapier. The sword was technically a secondary weapon to the musketeers, and in the book it specifically mentions that they are outfitted with their sword, two pistols, and a musket. At the end of the book we see the fab four fighting at the siege of La Rochelle. There they truely show off their marksmanship as they outperform all of the other members of their units. If you haven't read it then I highly recommend it. There are many good translations of the book, and many good audio versions. There was also a Russian TV series which was VERY close to the book, with only a few minor differences.