why basic character of 1^st group hydroxide increase down the group?

Basic character of hydroxide: The hydroxides of alkali metals behave as strong base due to their low ionization enthalpies, and so they can loose electrons and OH^- ions.

Due to the low I.P the M-O bond in M-O-H can easily break giving M⁺ and  OH^- ions. as; MOH → M⁺  +  OH^-  (basic nature).

As we move down the group from Li, Na to Cs, the Ionisation potential decrease and so it becomes more easier to loose electrons and hence the basic charatcter increases down the group.