Why was the memory of Tom Riddle not killed by the basilisk's glare? Even as a memory/horcrux, he is able to perform magic, and hence shouldn't he be affected by magic, magical objects and beasts too?
Was it that he was careful enough (like Harry) not to look at the basilisk at all the times he called it?
More generally, how does any wizard who use a basilisk survive it? Is it that they have its 'allegiance'?
There's no concept of "allegiance" discussed in canon.
It's plausible (though unproven - and as we see below, unnecessary) that Basilisk would avoid killing his "commander".
But the solution is even simpler: Basilisk kills only when you look into its eyes (NOT merely when it looks at you); so avoiding getting killed is simple - as you yourself noted, Harry managed to do it even when fighting it... so it would be trivially easy for Tom, who was off to the side and Basilisk's eye was turned to Harry.
I don't think a memory is truly alive (In regard to Tom Riddle, I mean.). Something that is not alive cannot be killed by any means. For example, Nearly Headless Nick¹ -- he couldn't die twice because he was already a ghost when he saw the Basilisk. Accordingly, the memory of Tom Riddle wasn't alive to be killed.
‘Justin must’ve seen the Basilisk through Nearly Headless Nick! Nick got the full blast of it but he couldn’t die again ... ’
Chamber of Secrets - page 215 - Bloomsbury - chapter sixteen, The Chamber of Secrets - Harry Potter
¹I used Nearly Headless Nick just as an example of a spectre-like being and how he couldn't be killed by the Basilisk; a memory isn't exactly a spectre, but it is non-corporeal (Unless it's made corporeal through emotion like Tom Riddle did with Ginny Weasley.). So, yes, I know Tom Riddle was never a ghost in Chamber of Secrets. It's just that a memory isn't corporeal or alive in the true sense of the word.
In any other sense you'd be correct. (If it can't kill a ghost how could it kill a memory?). But remember Nearly Headless Nick was petrified as a ghost, and Tom's memory, having gained corporeal form, was no longer just a memory. So if the basilisk can petrify a ghost with no physical body, what do you think it would do to a memory with one?
