Don't leave it up to someone else — take responsibility and do it yourself.
Literal
Others leaving-to not-doing, self [with-で] responsibility [object-を] holding do please.
彼方任せ (あちらまかせ, 'leaving it to others') is a compound where 彼方 means 'that side, the other party.' ~にしないで is the negative request ('don't make it into'). 自分で責任をもって ('taking responsibility yourself') stresses personal accountability. ~てください is the polite request. The sentence is a direct but polite admonishment.