“You must write for children the same way you write for adults, only better.”
Did he say or write this (or a Russian equivalent)?
Most probably not.
Besides Maxim Gorky this phrase is misattributed to S. Marshak and K.Chukovsky (both are famous Soviet children writers).
An answer to a very similar question (Russian text) provides a snippet from the book Цитаты из всемирной литературы от Гомера до наших дней (which translates as "Qutations of World Literature").
This snippet explains the author of the phrase is the son of Vasily Yan who used it in the preface to his father's 1969 book, Повести.
It explains it is a rephrasing of Stanislavski's phrase:
Для детей нужно играть так же, как для взрослых, но только - лучше
You must play for children the same way you play for adults, only better, more delicate, more cultured, more perfect.