汚い手で

with dirty hands

→ 汚い
phrase Rare #32,618
with dirty hands Here, で marks the means or manner of an action.
JA: 汚い手で料理しないで。
EN: Don't cook with dirty hands.

汚い手で私に触らないで

Don't touch me with your dirty hands.

— 箪笥<たんす> (2003)

汚い手で触らないで!

Don't touch it with your dirty hands!

— ロシアン・ブラザー (1997)

汚い手でこれに触らないで。

Don't touch this with dirty hands.

汚い手で私の本に触らないで。

Don't touch my book with your dirty hands.

そんな汚い手で触らないで。

Don't touch it with those filthy hands.

汚い手で触るんじゃない。

Don't touch it with your dirty hands.