That's almost correct. $ will match an optional newline as well, so both "5" and "5\n" are valid. Better to use \z

print "1. number\n" if "5" =~ /^\d+$/;
print "1. number and newline\n" if "5\n" =~ /^\d+$/;

print "2. number\n" if "5" =~ /^\d+\z/;
print "2. number and newline\n" if "5\n" =~ /^\d+\z/
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Update
The advice to use a \z fixes the problem of bad data being submitted, but it might block otherwise good data, depending on the input source. Data read from a filehandle could have the newline present, without it being from a misbehaving user.
chomping the field in question is a better approach than being stricter with the regex.