I can only speak for Oracle. A VARCHAR2(50) and a VARCHAR2(255) take up exactly the same amount of space and perform identically, if you enter the value 'SMITH'.
However, the reason why it is generally not a good idea to go around declaring all your textual columns as VARCHAR2(4000) is that column length is, effectively, another constraint. And constraints are database implementation of business rules, so they are definitely something that should be defined on the database side of things.
As a for-example. You define a CHECK constraint on a column so that the values it can accept are only 'Y' and 'N'. That saves your application from having to deal with 'y' and 'n' or even '1' and '0'. The check constraint ensures your data conforms to expected standards. Your application code can then make valid assumptions about the nature of the data it has to deal with.
Column length definition is in the same boat. You declare something to be a VARCHAR2(10) because you don't want it accepting an entry of 'ABC123ZYX456' (for whatever reason!)
In Australia, I define STATE columns to be a varchar2(3) because I don't want people typing in 'New South Wales' or 'South Australia'. The column definition pretty much forces them to be entered as 'NSW' and 'SA'. In that sense, a VARCHAR2(3) is almost as much a check constraint as actually specifying a CHECK IN ('NSW','SA','VIC' etc) constraint.
In short, proper column lengths are a way of encoding business rules. They're another form of constraint. They bring all the advantages of constraints (and suffer from many of the same drawbacks). And they ensure, to a small extent, a degree of 'data cleanliness' that "proper" constraints help with, too.
I don't buy the argument, either, that it's best to stick these sorts of things in the client app because it's easier to change there. You have 20,000 people using an app, that's 20,000 updates. You have one database, that's one update. The 'easier to change the client app' argument, if true, would potentially mean the database just gets treated as a giant bit bucket with all the clever logic being handled in client code. It's a big discussion to have, but since all RDBMSes let you define constraints and so on in the database itself, it's pretty clear that there's at least a worthwhile case to be made that such fundamental logic belongs in the backend.