List, map function based on a condition

Using ReplaceRepeated (//.) and pattern matching:

input //. {{x___, 0, 1, y___} :> {x, 0.5, 1, y}, {x___, 1, 0, y___} :> {x, 1, 0.5, y}}

{{0, 0, 0, 0.5, 1, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1}, {0.5, 1, 0.5, 1, 0.5, 0.5, 1, 0.5, 0, 0, 0, 0.5, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1}, {1, 1, 0.5, 0.5, 1, 1, 1, 0.5, 0, 0, 0, 0.5, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1}}

Note that there is a slight difference between this and your suggested output, highlighted in bold. As @Nasser received this result by his method as well, I suspect that the original suggested output was in error.

You can also use a combination of SequenceReplace and FixedPoint:

f = Map[SequenceReplace[{{0, 1} -> Sequence[.5, 1], {1, 0} -> Sequence[1, .5]}]], 

FixedPoint[f, input]

{{0, 0, 0, 0.5, 1, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1},
{0.5, 1, 0.5, 1, 0.5, 0.5, 1, 0.5, 0, 0, 0, 0.5, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1},
{1, 1, 0.5, 0.5, 1, 1, 1, 0.5, 0, 0, 0, 0.5, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1}}

You can also use Nest in place of FixedPoint:

Nest[f , input, 2] == %

True

For a rewriting problem, use rewriting explicitly:

input //. {
    {x___, 0, 1, y___} -> {x, 0.5, 1, y}, 
    {x___, 1, 0, y___} -> {x, 1, 0.5, y}
}
(*
    {{0, 0, 0, 0.5, 1, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1}, 
    {0.5, 1, 0.5, 1, 0.5, 0.5, 1, 0.5, 0, 0, 0, 0.5, 1,1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1}, 
    {1, 1, 0.5, 0.5, 1, 1, 1, 0.5, 0, 0, 0, 0.5, 1, 1, 0.5, 1, 0.5, 0, 0.5, 1, 0.5, 1}}
*)