● If you don't keep your promises/appointments, people won't take you seriously. ● People won't take you seriously if you break your promises.
「相手にしない」はnot take you seriously「真剣に受け止めない」で表現できます。「約束を守る」はkeep one's promises/appointments、「破る」はbreakを使います。
I asked for an appointment, but he wouldn't give me the time of day. 「アポを取ろうとしたが、彼は時間をくれなかった」。 He's the kind of guy who doesn't take women seriously. 「彼は女性をあまり相手にしないタイプなんだよ」。