The hydroxyl groups are hydrophyllic - that's what accounts for the solubility in water of glycerine and for its hygroscopic nature. I suspect it's also what's responsible for its ability to dissolve numerous inorganic salts. It's kind of like having a molecule that's half water, and half organics.
Glycerol forms a complex with the borax, forming monomeric or dimeric complexes with the glycerol. In borax, the sodium atoms are surrounded by borons atoms connected to other boron atoms with hydroxides bonded to the borons as well. The glycerol complex liberates som eof those boron atoms to form boric acid as it complexes. This free boric acid is what it lowering the pH of your solutions.
In fact, I once used this reaction as the basis to determine the amount of boron (I was looking for borax) in an industrial product that I was hired to deconstruct - I think there was a bit of industrial espionage going on with that project... Anyway, you can titrate a solution of borax (which is basic) with sodium hydroxide (also basic), using pH 7 as the endpoint.
Think about that for a second - you have a solution of borax with a pH of 10 or so, and you titrate it with sodium hydroxide with a pH of 13 or so - how do you get an endpoint of pH 7? (Isn't chemistry fun!?)
The simple way is to add an excess of mannitol (a sugar alcohol with a structure rather similar to glycerol) or glycerol to react with the borax. The borax releases boric acid as a result of the excess glycerol or mannitol. (Neither glycerol or mannitol have much effect on the pH of the solution.) You add some methyl orange or phenolphthalein indicators and titrate to the endpoint of the indicator. The amount of sodium hydroxide used in the titration is proportional to the amount of boric acid liberated in the complexing reaction. You can then calculate how much borax was present in the original sample.
I can list the chemical reaction equation if you like.
So I'd say by dissolving borax into glycerine, you are liberating boric acid and consuming your borax. Probably not what you are really trying to accomplish by using the glycerol as a "solvent".