I think the problem here is that we are only looking at a fragment of the problem and only waving arms towards Bayesian analysis without actually performing one. So lets try actually performing one.

Let B be the proposition that Linda is a banker, and L be the proposition that Linda is a librarian; furthermore let E (the evidence) be the event that Linda was seen walking into a bank.

Now, if Linda is a banker, then P(E|B) = 1 (bankers can reliably be spotted walking into a bank), but if she isn't then P(E|~B) = 2/10 (arbitrary value capturing the idea that banks have customers as well, who can also be spotted walking into banks, but less frequently). If we don't have any prior information about whether Linda is a banker, we might say P(B) = P(~B) = 1/2, however in practice we know that bankers are no were near as common as that, but it will do for an exploratory exercise. So lets plug that into Bayes' rule:

P(B|E) = P(E|B)P(P)/PE = P(E|B)P(B)/(P(E|B)P(B) + P(E|~B)P(~B)) = (1 x 1/2)/(1 x 1/2 + 2/10 x 1/2) = 5/6

So observing Linda walking into a bank does increase our belief that she is a banker from 1/2 to 5/6.

So what about librarians. In the original problem, we can have librarians that are also bankers, so we have a few more probabilities to set out. If Linda is a banker and a librarian, then P(E|B,L) = 1, if she is a banker but not a librarian, P(E|B,~L) = 1 (note P(E|B,L) = P(E|B,~L) = P(E|B) because B and L are independent). If she is a librarian but not a banker, or not a librarian or a banker, then she may still be a customer, so P(E|~B,L) = P(E|~B,~L) = 2/10 (note P(E|~B,L) = P(E|~B,~L) = P(E|~B) as B and L are independent). Similarly, for the prior, we have P(B,L), P(B,~L), P(~B,L) and P(~B,~L). To keep life simple, lets make them all equiprobable, i.e. 1/4.

Lets run that all through Bayes' rule:

P(B,L|E) = P(E|B,L)P(B,L)/P(E)

Lets work out P(E) first because we don't have LaTeX on this forum.

P(E) = P(E|B,L)P(B,L) + P(E|B,~L)P(B,~L) + P(E|~B,L)P(~B,L) + P(E|~B,~L)P(~B,~L)

P(E) = 1/4*(1 + 1 + 2/10 + 2/10) = 1/4 x 24/10 = 24/40 = 3/5

So

P(B,L|E) = P(E|B,L)P(B,L)/P(E) = (1x1/4)/(3/5) = 1/4 x 5/3 = 5/12

and

P(~B,L|E) = P(E|~B,L)P(~B,L)/P(E) = (2/10 x 1/4)/(3/5) = 1/4 x 1/3 = 1/12

Marginalising

P(L|E) = P(B,L|E) + P(~B,L|E) = 5/12 + 1/12 = 1/2

In other words, observing Linda walking into a bank raises the probability that she is a librarian from the prior probability of P(L) = P(B,L) + P(~B,L) = 1/4 + 1/4 = 1/2, to, err... 1/2! It tells us precisely nothing about whether Linda is a librarian.