It's ok to own a dog, but make sure to raise it responsibly.
「責任を持って育てなさい」は英訳すれば、「Make sure to raise it responsibly.」になります。命令するとき、「make sure to」という表現をよく使います。
「raise it」は「(犬を)育つ」という意味です。「responsibly」は「責任持つ」を表しますがこの場合で副詞として使っています。
「飼っても良いけど、責任持って自分で育てなさい」も翻訳すれば「It's ok to own a dog, but make sure to raise it responsibly.」になります。
「It's ok to own a dog, but」は「(犬を)飼っても良いけど」という意味です。
You can keep it, but you have to take responsibility for it.
You can keep it, but you have to take responsibility for it.
If you’re going to keep it, you need to take full responsibility.
You can have it, but you’re the one who has to look after it.
If you want to keep it, you have to raise it yourself.
You can keep it, but you’re responsible for taking care of it.
「責任を持って育てなさい」は、英語ではよく “take responsibility” や “be responsible for” を使って言います。