In all the shower installs I overseen, the company I worked for maintained that caulk is to be used in any inside corners except where excessive/standing water is. For example, caulk corners where tub and walls meet, and vertical inside corners where the walls come together. DO NOT caulk where the floor and walls meet, I personally seen caulk creep out of the joint after a week of use in a residence I worked on while I was finishing up the punch list.
Find a reputed mold/mildewcide to clean the joint after you clear all the old grout then allow to dry as long as possible and use a color matched caulk. This is available at Home Depot or at tile supply houses. It is available in sanded or unsanded.
In theory, it may trap moisture, but when the tile is installed and the grout sealed, which should be the case, between the tile being water impervious and the grout has been sealed from moisture infiltration, the amount of water getting past both those points is very minor in my opinion, I read this is so elsewhere.
11-30-2013 addition
To clean the joint of old grout, protect the tub with blue painters tape, maybe a few layers and use a razor knife to begin to V cut the grout out, again be careful the side of the razor knife will be rubbing very hard on the tub/tape and can wear through readily. There is a grout saw available at hardware stores, but they will cut a joint about 1/8" wide, most tile is set closer. If you use this, use a metal shield of some sort to protect the surface of the tub. A large drywall knife (12") would work or a section of very thin gauge metal purchased from a hardware store where they sell small pieces of flat metal. Careful with this, it must be kept secure from moving or the metal will deposit gray marks on the acrylic. Tape it to the tub at the ends and let the metal go up to the face of the tile so any scratches is covered by the new caulk.
To caulk, first the joint MUST be clean and dry, cut the tip to the size of the joint, and apply caulk. Look as the caulk is exiting the tip to ensure the caulk is going INTO the joint, this is where the work is done by the caulk, This why it must be clean. Yes there will be caulk EVERYWHERE to a degree. This is why I keep a 5 gal. bucket handy half filled with water and a rounded edge sponge to clean the surface. Paper towels won't last in the cleaning process, and it may gouge out the caulk if a folded corner of the paper towel goes awry.
To clean the excess caulk, with the wetted sponge, squeezed the daylights out of it to get most of the water out of it. It is imperative you are not depositing too much water on the surface of the tile, it will weaken the caulk. You can clean the caulk from the surface completely leaving the caulk only in the joint and with the color matched caulk, it will look like the other joints on the wall. These caulks are specified to tubs and showers.
That should have never been grout, joints between walls and other objects should be caulked because movement in those joints will cause the cracking that you now see, and the cracking can lead to water getting behind the tile. This also goes for inside corners in a shower, and the joint between bottom of the wall and the floor or tub..
Best Answer
OK, grout should never have been used there in the first place because it will crack. I can see more cracking from your pictures. Your best bet is get as much of that grout out of there as you can as it will continue to crack and fall out. Once you get the grout out, get some quality silicone caulk and run a good bead of caulk around the tub. Make sure everything is dry before caulking.