Stay woke is an urban idiom that implies stay informed to the truth in the midst of biased, filtered information…Don’t get hypnotized by current trends, pop culture or status quo…Don’t let it dictate your thinking. It can also infer don’t believe your own hype.
That being said, from a Christian point of view it simply means don’t get caught up in the world.
I can’t say it any better than how God said it through the Apostle Paul…
Romans 13:11-14 (NLT)
11 This is all the more urgent, for you know how late it is; time is running out. WAKE UP, for our salvation is nearer now than when we first believed. 12 The night is almost gone; the day of salvation will soon be here. So remove your dark deeds like dirty clothes, and put on the shining armor of right living. 13 Because we belong to the day, we must live decent lives for all to see. Don’t participate in the darkness of wild parties and drunkenness, or in sexual promiscuity and immoral living, or in quarreling and jealousy.14 Instead, clothe yourself with the presence of the Lord Jesus Christ. And don’t let yourself think about ways to indulge your evil desires.