This is an interesting problem. I can think of two mechanical (brute-force-ish) approximate approaches. My math-fu is not strong enough to opine if an analytic solution is practical here. I hope there is such an approach! But here’s my “just get it done” suggestions.
By Gridded Area, approximate
We only care about the blue area. Represent that to the desired resolution, as a 2 dimensional array of “blue” or “not blue”, and start it with your circle. Then knock out red circles one by one. Any blue left is not in red (within the grid resolution).
Improvement 1. skip red circles that don’t intersect, dist(center1,center2) > (radius1 + radius2).
Improvement 2. instead of just Blue or Not-Blue, store a “distance from red center”, and when knocking red areas out, also fill in the whole blue area with the minimum distance to any of the red centers. Now the highest remaining blue cell is the “least crowded”.
Improvement 3. Keep the blue cells in a linked list, and knock out ones that are in red. For each new red circle, just walk the still-living blue cells.
Could work in 3d, but the resolution is even more expensive. The spheres would be sets of cubes, minecraft-style.
By Polygon, approximate
Represent your circles by N-gons, with whatever number of sides is “good enough”. Use boolean operations to carve away the blue area. Remaining blue polygon is blue, but it might have no sides left, meaning no safe blue area left. http://en.wikipedia.org/wiki/Boolean_operations_on_polygons has some notes on this, and links to boolean-operation libraries.
Could work in 3d, but, those boolean ops are even more expensive.
By Spring Forces
If the red circles/spheres are not necessarily absolutely solid, or if they represent zones of danger, another approach would be to apply forces of various kinds. If your companion object A tries always to move towards B, but is forced away by R, it will tend to find a way to get to B. This will work best if everything is moving, since "unsolvable" positions may get another chance. Something nonlinear pushing away from the the R circles, from gentle nudges far away, to "impenetrable" when A is at their actual physical boundary, would be appropriate.
If this is for game collisions, the grid or polygons may be good enough, especially for a low resolution or low polygon sides. (If this were an abstract math question… my answers would be unsatisfying.)
For a game AI, though, perhaps some variation on the "spring forces" might serve better, giving the object some lively and persistent motions. The simplest force would have your "companion" object A just follow a greedy path, and aim straight for B until it hits any of R, and then stop or slide around the Rs.