How to handle Part function error when position does not exist?

But what if I wish to handle that error. So in this case, I want to return empty list {} if the part-does-not-exist error happens

One possibility might be

ClearAll[process];

process[lis_List, (n_Integer)?Positive] := Module[{},
       Quiet@Check[lis[[n]], {}, Part::partw]
]

Or just

process[lis_List, (n_Integer)?Positive] := Quiet@Check[lis[[n]], {}, Part::partw]

And now

list = {{1, 2}, {3, 4}};
process[list, 2]
 (*{3, 4}*)

and

process[list, 3]
(* {} *)

Quiet can be removed if you want to hear the beep and see the error message on console.

As with the essence of @Nasser's answer, I go with a simplified version for now.

Quiet[Check[{{1, 2}, {3, 4}}[[3]], {}], Part::partw]

This returns empty list if partw message happens.

Have you considered Query? Default behavior:

list = {{1, 2}, {3, 4}}

Query[3][list]

Missing["PartAbsent", 3]

Somewhat customized:

fn = Replace[Missing["PartAbsent", _] -> {}];

Query[3, fn][list]

{}

Or with care:

Unprotect[Missing];
Missing["PartAbsent", _] = {};
Protect[Missing];

Query[3][list]

{}

Caveat, Query can be slow:

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