I think you are misinterpreting what the BioLogos people are saying.
Under "What We Believe":
http://biologos.org/about
* "We believe that God created the universe, the earth, and all life over billions of years. God continues to providentially sustain the natural world, and the cosmos continues to declare the glory of God. Therefore, we reject ideologies such as Deism that claim the universe is self-sustaining, that God is no longer active in the natural world, or that God is not active in human history."
* "We believe that God continues to be directly involved in human history in acts of salvation, personal transformation, and answers to prayer."
* "We believe that the diversity and interrelation of all life on earth are best explained by the God-ordained process of evolution and common descent. Thus, evolution is not in opposition to God, but a means by which God providentially achieves his purposes. Therefore, we reject ideologies such as Darwinism and Evolutionism that claim that evolution is a purposeless process or that evolution replaces God."
So it is not accurate to say that "BioLogos-style" theistic evolution includes the belief that "God didn't intervene in the history of evolution" ... in fact precisely the opposite!
The BioLogos position is that the unfolding of evolution *IS* God's method of creation.
So it is as meaningless to ask how God "intervenes" in that process as it is to ask how a painter "intervenes" in his own process of painting ... or to ask how you "intervene" in your own process of creating thoughts.
As for the answering of prayers (which, as you can see, is also part of the BioLogos statement of belief), this is also complex.
One essay on the BioLogos foundation that addresses this specific point, is this one by Paul Lange. As a cancer surgeon, he describes the challenge of being a medical doctor who deals with the human body as a naturalistic organism, at the same time he deals with his own faith, and the faith of those praying for miracles.
"Does God Change His Mind?"
http://biologos.org/blog/does-god-change-his-mind
In his view, once one accepts the natural world as not being separate from ... or a static creation of ... God, but rather something so inseparable from God that He is in a *constant* state of involvement with it ... then divine action, including in response to prayer, is just a part of that same process. Answering prayers is not an "exception" to his normal involvement with the natural world, it is absolutely consistent with it.
"As a scientist it is much easier to believe in DA [divine action] than it was when the worldview was so-called static; that is, the view that the world was created in toto, once and for all, and governed by infallible consistent “laws.” Now we “see” a much different universe; one that started with the big bang, has a history and geography that positions our world and a humankind (that evolved over eons of time) into a very small place and time; black holes; dark matter; time travel; string theory with unfathomably dimensions exceeding six; maybe multiple universes making other intelligent life in the cosmos probable; and so forth. Taken together this “ real” world is now much more unfathomable than Eve from a rib, water into wine, or a resurrection. No wonder physicists concerned with matters related to cosmology are now so filled with wonder that they appear spiritual."
Please don't do what the creationists do ... and try to reduce things down to a cartoon.