PostgreSQL check if array contains any element from left-hand array

Assuming that your inputs are arrays but it is okay to unwrap them using unnest(), here is a solution:

SELECT count(*)>0
FROM
    (SELECT unnest('{2,3}'::int[]) a1) t1
    join (SELECT unnest('{1,3,4,7}'::int[]) a2) t2
        on t1.a1=t2.a2;

Sure, use the && array-overlaps operator:

SELECT ARRAY[1,2] && ARRAY[1,3,4,7];

See array functions and operators.