How do "basic" amino acids (lysine, arginine) act as base under physiological conditions? Why do we consider amino acids with high side chain pKa's basic, yet with a higher pKa, less of the amino acid will exist in basic form at equilibrium compared to acid form at physiological pH?

What will the concentration of the base be given the high pKa of these amino acid side chians. The answer is that the higher the pKa, the lower the concentration of the base, since with 1) with a higher pKa, then 2) you will have a lower Ka (pKa = - logKa) and therefore 3) a lower concentration of base, due to Ka = the conjugate base (times H3O+) / acid, then the numerator, the conjugate base will be  lowered in concentration with a lower Ka. This can also be confirmed with the Henderson-Hasselbach equation, where pH = pKa + log [A-]/[HA] and since pkA is increasing will make pKa > pH, meaning pH - pKa = a negative number = log [A-]/[HA], where [A-]/[HA] would then equal 10^negative number, meaning the concentration of the acid will be greater than the concentration of the base.

My question is, when you look at titration curves, we know that parts of an amino acid will act more as a base, than an acid, at higher pKa's. Meaning, if you have the amine group of an amino acid, with a pkA of 9-10, it doesn't want to give up it's hydrogen, until the pH is pretty high (10-12). Why are amino acids with high overall pKas considered basic, yet, when you look at a molecule's pKa at a high pKa, less of the molecule will exist as the base at equilibrium (meaning it is wrong to say that a molecule with a high pKA will exist primarily it is basic form)?
I will paraphrase the question:


  Why are the amino acids arginine and lysine called basic? Why are the amino acids glutamic acid and aspartic acid called acidic?


This nomenclature makes sense for situations where the amino acids are dissolved in a non-aqueous solvent. In that case, the side chains will be neutral and have nitrogenous base groups and carboxylic acid groups, respectively. 

When the amino acids are dissolved in water at neutral or physiological pH, however, the name does not make sense because they have already acted as base (Arg and Lys) or as acid (Glu and Asp), and are now the conjugate acid and conjugate base, respectively. For glutamic and aspartic acid, it is common to refer to them as glutamate and aspartate (the name of the conjugate base). For arginine and lysine, even when protonated, those names are usually used nonetheless.


  [OP in comments:] So, in physiological situations (aka in the human body), how do arginine, lysine, and histidine act as bases, if they exist in their protonated form, considering how they are often used in enzymatic catalysis (same question asked of acidic amino acids in their deprotonated forms)?


Histidine

Histidine is a common acid/base catalyst. Its pKa is close to 7, so an appreciable fraction is protonated and an appreciable fraction is deprotonated. Often, this amino acid will act as acid and as base in different steps of the same mechanism (e.g. RNase).

Lysine

Lysine is not a typical acid base catalyst. It has roles in binding negatively charged ligands (such as ATP), acting as a "positive charge on a stick". It also acts as a nucleophile, forming e.g. Schiff's base with aldehydes and ketones. Lysine's pKa in the context of a folded protein varies considerably (see e.g. https://www.pnas.org/content/108/13/5260), so in some contexts, it will be an efficient acid/base catalyst.

Arginine

Arginine is not a typical acid base catalyst. Like lysine it has roles in binding negatively charged ligands but different from lysine, it can act as a bidendate hydrogen bond donor (e.g. making a nice salt bridge with aspartate or glutamate with two hydrogen bonds). It also has a role in cation-pi interactions, stacking its planar guanidinium group on aromatic rings. This web page has more on the functional role of arginine: http://www.russelllab.org/aas/Arg.html (and there are similar pages for the other amino acids on that site)