As with all "use" directives, defining a constant happens at compile time, before the code is run. At compile time, $Base has no value. That is why this works:

#!/usr/bin/perl -w
use strict;

use constant BASE => 'mybase';
print "BASE: ", BASE, "\n";
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and your code does not.

Update: If you want to set a 'symbolic constant' to a variable, you could use a variable and stick to the convetion that all caps means constant:

!/usr/bin/perl -w
use strict;

my $BASE = 'mybase';
print "BASE: $BASE\n";
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This also has the advantage of allowing variable interpolation.