Firefox's tabbed browsing feature encourages breadth-first search. If I click on a link the tab containing the page linked to appears as the rightmost tab, and I generally work through tabs from left to right. Breadth-first search can be implemented in this way -- we maintain a list of pages to be looked at, and the first page to enter the queue is also the first page to leave it.
Depth-first browsing wouldn't be too much different on a cosmetic level -- it could be set up by having the new tab appear immediately after the current tab, giving a stack of pages to view instead of a queue. I suspect the subjective experience of Internet browsing would feel much different from such a point of view -- browsing often seems to lead to shallow knowledge. (If one had time to search the entire Internet breadth-first search and depth-first search would eventually visit the same set of pages -- but who has that kind of time?) The "optimal" algorithm for finding particular information isn't strictly breadth-first or depth-first, though; if you think about how you search when you look for a specific piece of information, you don't routinely follow the leftmost tab or the rightmost tab, but instead click on whatever tab subjectively seems like it would give the best information.