Would your solution to this be have huge if/elsif statement . . .

No, I'd put the columns I wanted for each table in an HoA:

my %COLUMNS = (
    MY_FIRST_TABLE => [ qw( blah_1 blah_2 ) ], 
    MY_2ND_TABLE   => [ qw( another_col dob ) ], 
    . . . 
);

# $table defined elsewhere
my $SQL = 'SELECT ' . join(',', @{ $COLUMNS{$table} }) . " FROM $table
+";
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Clean, efficient, no SELECT *.

Update: Added some missing whitespace in the SQL generated.

. . . describe table (which BTW is not portable to other DBMSs) . . .

MySQL's describe table isn't, but DBI's column_info method is. (Though I wouldn't use that particular solution).