I always like to build or fashion some sort of spacer made from treated wood to be on the concrete but between the concrete and the actual posts.
This spacer or 2x6 can be glued down with liquid nails for concrete and do the job for you. then use clear 100% silicon to caulk around the spacer to keep water from under the spacer.
Then get four "L" brackets, found at any home center and then fasten them to the top of the spacer and to the sides of the post base. Use stainless steel screws.
Or, drill a 1/2 inch hole in the center of the spacer and using a lag bolt fasten the spacer to the base of the post. Then mark your intended spacer for the post and liquid nails that section and hammer it into place.
What I always do when replacing posts is buy a 4x4 post and temp place it in the close area where the old post will go. Cut the 4x4 about 1/2 inch taller than normal and hammer it carefully up right. Now the old post annd the spacer can be placed into the slot. Make the spacer and the post about 3/8 inch taller than the space, then knock out the 4x4 carefully and the weight on the posts will also keep it snug in place.
If you forgoe the spacer then you can still use the "L" brackets and just drill into the concrete and use anchors for concrete and your set. sometimes drilling into old concrete is one challenge in itself and will use many bits and the bit starts "walking" and the holes get off set.
Use a titanium concrete drill bit. Cost four times the price of the cheap ones, but will do the job first time and every time.
This is not a difficult project by any means, just so many ways of doing it. In my area, the treated wood is definately the route to go. Our foundations move so much the fastening of the "L" bracket to the concrete to the posts will not last at all.