Has seemingly to do with argument $k r$:

In[5]:= Integrate[BesselK[1, k r] BesselI[1, k r], {r, 1/2, 3/2}, Assumptions -> k \[Element] Complexes]
Out[5]= Integrate[BesselI[1, k r] BesselK[1, k r], {r, 1/2, 3/2}, Assumptions -> k \[Element] Complexes]

stating that $k \in \mathbb{C}$ does not help, so substitute the argument

    In[7]:= Integrate[BesselK[1, x] BesselI[1, x]/k, {x, k/2, 3 k/2}, 
                      Assumptions -> k \[Element] Complexes]

    Out[7]= (1/(4 k Sqrt[\[Pi]]))
            (-MeijerG[{{1, 1}, {}}, {{1/2, 3/2}, {-(1/2), 0}}, k/2, 1/2] + 
              MeijerG[{{1, 1}, {}}, {{1/2, 3/2}, {-(1/2), 0}}, (3 k)/2, 1/2]
            )

And here is the latest and greatest

In[4]:= $Version
Out[4]= "11.3.0 for Microsoft Windows (64-bit) (March 7, 2018)"

In[5]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, 1/2, 3/2}]
Out[5]= 2/9 (8 + \[Pi]^(
    3/2) (9 MeijerG[{{1/2, 1}, {3/4, 5/4, 2}}, {{1, 1, 1}, {0, 0, 3/4,
           5/4}}, 1/2, 1/2] - 
      MeijerG[{{1/2, 1}, {3/4, 5/4, 2}}, {{1, 1, 1}, {0, 0, 3/4, 5/
         4}}, 3/2, 1/2]))

In[6]:= % // N
Out[6]= 1.45921

In[7]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, .5, 1.5}]
Out[7]= 1.45921

Can you explain this?

It's a neverending story to explain software (implementations). Nevertheless Mathematica has an overwhelming strong relation to science and there it's quite usual to switch to numerical results (in contrast to symbolic ones) if the user chooses to give numerical input. Compare

In[8]:= 1/2
Out[8]= 1/2

In[9]:= 1/2.
Out[9]= 0.5

nothing obscure in that.

Hi Dent de Lion,

I really appreciate your investigation of my problem. You are correct that they argument $kr$ seems to be to blame for some of these computation failures. However, even with a simple argument, Integrate acts differently for float limits than for rational limits. Can you explain this?

For example,

In[1]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, 1/2, 3/2}] // InputForm

Out[1]//InputForm=Integrate[(BesselI[0, x]*BesselK[1, x])/x^2, {x, 1/2, 3/2}]

In[2]:= Integrate[BesselI[0, x] BesselK[1, x]/x^2, {x, .5, 1.5}]

Out[2]= 1.45921

Thanks, Jeff