R' Moshe Feinstein addresses your question (Igros Moshe YD I §54). He says that they are not believed under the category of eid echad ne'eman b'issurin (עד אחד נאמן באיסורין - "a single witness is believed to relate whether something is a forbidden item")1 since they don't keep kosher.
However, he also says that if you know from extensive experience that a particular close family member absolutely would never lie (or at least you know that they absolutely and certainly would not lie to you) you can believe (under pressing circumstances) what they tell you about whether they cooked kosher food or whether they used kosher utensils, due to the principle of kim leih b'gaveih (קים ליה בגויה, see K'subos 85a).2 3
This principle would not apply broadly to anyone who is non-observant but has a reputation for honesty, because the principle of kim leih b'gaveih only applies to people with whom you have such extensive experience that you know for certain that they would not lie to you under any circumstances (Igros Moshe YD II §43). Even Rava's righteous student Rav Papa did not meet this threshold with respect to Rava.
See this article (in Hebrew) for more discussion on the topic of believing people about kashrus if they don't keep kosher.
1 Discussed in this answer.
2 The gemara over there relates that Rava accepted the word of his wife in a court case, but he would not accept the word of Rav Papa as a solitary witness. This is because Rava had extensive personal experience with the absolute reliability of his wife's word, while he did not have the same degree of personal experience with Rav Papa.
3 The person preparing the food should also be sufficiently aware of the laws of kashrus (e.g. they know which utensils are kosher, they know which kosher symbols are reliable, they know not to mix any non-kosher food or utensils with the kosher ones, they know how to properly separate dairy and meat, etc.) so that they do not feed you non-kosher food in ignorance.