What level of attachment to one’s family is acceptable?

The acceptable level of attachment is that which does not cause us to disobey Allah. This is not something which can be defined in precise terms, but can be defined in terms of what it is not. One can love one’s family to do things for them but if that love or obedience leads us to disobey Allah and to do what is displeasing to Him, then this is unacceptable.

Allah says in the Qur’an:

يَـٰٓأَيُّہَا ٱلَّذِينَ ءَامَنُواْ لَا تُلۡهِكُمۡ أَمۡوَٲلُكُمۡ وَلَآ أَوۡلَـٰدُڪُمۡ عَن ذِڪۡرِ ٱللَّهِ‌ۚ وَمَن يَفۡعَلۡ ذَٲلِكَ فَأُوْلَـٰٓٮِٕكَ هُمُ ٱلۡخَـٰسِرُونَ

“O you who have believed, let not your wealth and your children divert you from the remembrance of Allah. And whoever does that - then those are the losers.” [1]

Reference:

[1] Surah al-munafiqun, 63:9