To an extent, yes it does have negative connotations.
The word was first used in cycling as an insult for Maurice Brocco, known as Coco, in 1911. Brocco started six Tours de France between 1908 and 1914, finished none of them, although a stage he won in 1911 caused the coining of domestique.
In road racing, if you are young and unproven you often get your start as a "domestique" on a team. You are the cycling equivalent of the chore boy. The job of the domestique is to sacrifice his or herself for the leader and team.
If one of the stars an your team needs some food, you have slip out of the peloton, and go back to the support cars pick up the food, then work your way back up to and through the peloton to deliver the goods. Likewise if you are in a situation where either the domestique or the star gets water, the star wins out.
If the team is worried about a break getting away, it will often fall to the domestique to set the pace and do all the work to close the gap.
Basically, the domestique is an expendable asset of the team. Their final race ranking is of little to no concern, other than to remain in the race so that they can provide their services.
That said, as you prove yourself as a domestique (ability to ride hard and survive) you will be given more opportunities, eventually working your way up in the team. If you are a fan of the sport, following the domestiques may give you an idea of who may be the upcoming stars.