When it comes to bond energy or molecular binding energy, do people mean enthalpy?
When in doubt, consult IUPAC Gold Book for definitions (italics below is mine).



bond-dissociation energy, $D$

The enthalpy (per mole) required to break a given bond of some specific molecular entity by homolysis, e.g. for $\ce{CH4 -&gt; CH3^{.} + H^{.}}$, symbolized as  $D(\ce{CH3−H})$.



bond energy (mean bond energy)

The average value of the gas-phase bond dissociation energies (usually at a temperature of  298 K) for all bonds of the same type within the same chemical species. The mean bond energy for methane, for example, is one-fourth the enthalpy of reaction for: $$ \ce{CH4_{(g)} -&gt; C_{(g)} + 4H_{(g)}} \, .$$ Tabulated bond energies are generally values of bond energies averaged over a number of selected typical chemical species containing that type of bond.



So, yes, conceptually you should use enthalpy. Note, however, that the enthalpy of a molecular specie in gas phase at, say 298K, is calculated as follows 
$$H_{298} = E_{\mathrm{e}} + E_\mathrm{ZPE} + H_{\mathrm{trans}} + H_{\mathrm{rot}} + H_{\mathrm{vib}} + RT \, ,$$
where 


$E_{\mathrm{e}}$ is the electronic energy;
$E_\mathrm{ZPE}$ is the zero point energy;
$H_{\mathrm{trans}}$, $H_{\mathrm{rot}}$ and $H_{\mathrm{vib}}$ are the temperature-dependent contributions to enthalpy from translation, rotational and vibrational motion, respectively.


And the problem (as Martin mentioned in comments) is that basically only the first term out of this five can be calculated accurately enough when the ultimate goal is to calculate energies (bond energies in our case) with the so-called "chemical accuracy" which is about $\pm 1$ kcal/mol.
For relatively large molecules harmonic frequencies and consequently $E_\mathrm{ZPE}$, as well as a rigid-rotor harmonic-oscillator approximation used to calculate thermal corrections, become inaccurate, and thus, the change of electronic energy should always be used as a reference.

$%edit$