Neutralising sulfuric acid but not boric acid
You've got to know that sulfuric acid is a strong acid and boric acid is a very weak acid. So, sulfuric acid can neutralize most of the bases where boric acid cannot. So, you have to take a base that can be neutralized by only sulfuric acid and not boric acid. So, you should probably take group 2 hydroxides and try to react with the unreacted sulfuric acid because sulfuric acid reacts with group 2 hydroxides whereas boric acid can't.

$$\ce{M(OH)2 +  H2SO4 -&gt; M2SO4 + 2H2O}$$

$$\ce{M(OH)2 +  H3BO3 -&gt;nothing}$$

where $\ce{M=Be,Mg,Ca,Sr,Ba}$.

So, which group 2 hydroxide to use? Group 2 hydroxide react with the sulfuric acid to form group 2 sulfates. Now, solubility of group 2 sulfates decreases down the group. So, you should use calcium hydroxide so the calcium sulfate that forms will not dissolve in the solution due to its low solubility and form a precipitate. You can separate that precipitate by filtration. 

Formation of calcium borate is highly unlikely and so only calcium sulfate will only form.

Also, barium hydroxide is the best chemical for this process because barium sulfate is very insoluble. But barium hydroxide is costly if you buy it or making it is difficult due to the cost of other reagents. Instead, calcium hydroxide is your best bet.