How do "basic" amino acids (lysine, arginine) act as base under physiological conditions?
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)