Were costumes important at the time of Shakespeare's Globe theater?
Yes, costumes were very important! However, the Elizabethans didn't think of costumes in the literal sense (as helping to creating the time and place of the action of the play) that we do today.
Our theatrical conventions (the things that audiences expect to see onstage) in the world of modern theatre are quite different from the conventions of English Renaissance theatre. Today, as theatre audience members, we expect costumes onstage to accurately represent the time and place of the play we are watching. We look for details in sets and props to support whatever time/place are suggested by the costumes, so that they send a unified picture of the world of the play. Is it an Elizabethan staging? Well, then we better not see any blue jeans or cell phones onstage. Is it set in the 1940s of World War II? Then no sword fights or pumpkin pants and hose, please!
Elizabethan theatre conventions weren't quite as clear-cut. Shakespeare's actors wore the dress that was "modern" at the exact time the plays were staged. So, for example, Julius Caesar, Mark Antony, Cleopatra, Hamlet, Claudius, Macbeth and King Lear would have all worn costumes signifying their important (and exalted) places in the upper crust of society. But all of these characters would have worn modern Elizabethan dress.
The costume was meant to help the audience distinguish between the social status of different characters, not to signify time and place. Certainly, there might have been a nod to the toga-style dress of the Romans, but this would have been far over-shadowed by the elaborate Elizabethan costuming meant to signifying the important of characters as understood by the Elizabethans view of their own world.
This may seem confusing or incorrect to us today, but there was nothing literal about the world of theatre during the Renaissance. The idea that events onstage are supposed to replicate "real life" and that the audience should remain invisible before the world created by the actors onstage would have seemed a ridiculous notion to the Elizabethans. Their sense of theatre wasn't literal at all. The Globe used hardly any set pieces at all, and the costumes were drawn from clothes readily available and recognizable to the audiences who attended the performances.
For more on theatrical conventions of the English Renaissance, please follow the links below.