What happens to the thickness of boundary layers as you speed up? I recently watched a video about the SR-71, in which it said “the inlet spike develops a significant boundary layer of air” when talking about the aircraft at Mach 0.5.
I thought boundary layers got thinner as you sped up, but is it different at such a high speed? The video is here, and it says the things about the BL at around timestamp 4:30.
There are different measures of boundary layer thickness. The Wikipedia article goes through them.
You'll see that they grow with length along the body, but get smaller with Reynolds number (which increases along the body). These effects somewhat counteract, preventing linear growth of the boundary layer along the body. Overall, it does grow along the body.
However, the freestream velocity part is with the Reynolds number in the denominator. I.e. At higher speed, the boundary layer gets thinner.
Mach 0.5 is slow for an SR-71.
Engines are particularly sensitive to ingesting boundary layer air. First, ingesting a boundary layer means that the flow entering the engine is non-uniform. This is more challenging for the turbomachinery -- you can't run as high of pressure ratios because some of the flow will have a different angle than other areas and you will stall the blades. Consequently, you must back off of what you might have been able to operate at any time you have nonuniform inflow. Secondly, the air in the boundary layer has had a total pressure loss due to the viscosity. This means that air will be less efficient in the engine.
Most supersonic aircraft have some sort of boundary layer diverter to avoid ingesting the boundary layer from the forward fuselage into the engine. Even with that, some also have boundary layer suction in their inlet ramps.
As you have commented -
Oh okay, so one follow up question: what exactly makes the Reynolds number increase along a body? The flow would get more turbulent, so does that have something to do with it?
The principal controlling factors regarding Reynolds number, are a) velocity of the flow over the surface, and 2) length of the flow path over that surface. As you may recall, another question that you asked was in regard to turbulence. But along that line, we must also consider development of the laminar layer and the extent of the laminar boundary layer as the flow progresses over the surface in the downstream direction. A favorable pressure gradient will enhance the presence of the laminar boundary layer and, remembering that as the object's translational velocity increases, the Reynolds number will increase. Therefore, as a consequence, presence of the laminar boundary layer will be enhanced. This can occur for the laminar boundary layer up to, and in excess of, a Reynolds number of 60 million! For this to occur, perfection in the surface contour and curvature is paramount.
So what about the turbulent boundary layer? Several factors come into play because the turbulent boundary layer forms where the flow is progressing against an adverse pressure gradient. Factors regarding the growth and continued attachment of that boundary layer are related to the shape, i.e. concavity, of that surface, and surface condition, i.e. smoothness and curvature, of that surface. In another answer to a question regarding the turbulent boundary layer resulting from surface contamination on the leading edge of an airfoil, the following was noted regarding a comment by FX Wortmann -
Wortmann makes note of a rather startling fact: 'If we compare the roughness height, which will not shift the transition and not increase skin friction, we find for higher Reynolds numbers that the turbulent boundary layer requires usually a smoother surface than the laminar flow…’
Information regarding these aspects of fluid mechanics is readily available in the literature. Many free resources are readily available on the internet. Try finding a few of them and take a look. Reading about this will help you understand the different aspects of what is going on. And if you have the time, a suggestion would be to do the following: Either in person, or via the internet, take an up-close look at the condition of a modern aircraft and note the perfection presented in its surface construction. A resource for this examination was provided in your last question regarding why boundary layers become more turbulent as they flow over a surface. 
A Gulfstream G-800. Note the perfection in the surface smoothness of the wing as evidenced by the reflection of the fuselage, engine nacelle, and tail.
