Oddly enough, I had a requirement to do the same yesterday, and found that this works:

use DBI;
use strict;

my $db=DBI->connect("DBI:SQLite:dbname=testdb","","");

my $sth = $db->prepare("select * from nosuchtable");

if ($sth){print "Table exists\n"}
else{print "No such table\n"}
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(i.e. without doing an execute).

The code displays a warning from SQLite that the table doesn't exist, and the true test for $sth fails. Of course, this might be a bad way (if it is, I'd like to know so I can change it in my code).