How do I know which molecular structure is glycine's "base state"?
Short Answer: You need to define base state.

Background: The point to keep in mind here is that the base state (as you call it) is not constant. As @jeppenielsen points out in the comments, amino acids are zwitterions. According to Wikipedia:


  In chemistry, a zwitterion, formerly called a dipolar ion, is a neutral molecule with both positive and negative electrical charges. (In some cases multiple positive and negative charges may be present.) Zwitterions are sometimes called inner salts. Unlike simple amphoteric compounds that may only form either a cationic or anionic species, a zwitterion simultaneously has both ionic states.


In glycine, the center of positive charge is the -NH2 group while the center of negative charge is the -COOH group. At different pH (acidic, neutral or basic), the structure of the molecule keeps on changing as shown in the image below:

source

Structure of other amino acids also changes similarly. To know why this change occurs, just keep in mind that (in very simple terms) pH is inversely proportional to the concentration of H+ ions in a solution (higher the pH, lower the concentration of H+). The exact formula is $\ce{pH = - log[H_3O^+]}$ (yes, it means that a pH of 4 is 10 times more acidic than pH 5 and so on). Also, as pH changes, so does the formula for the amino group (see the diagram again).