According to Vol. I of Bateman (p. 196 in my copy),
$$ \int\limits_0^{+\infty} t^\mu I_\nu(at)\,e^{-pt} dt = \Gamma(\mu+\nu+1) \, s^{-\mu-1} \, P_\mu ^{-\nu}(p/s), $$ where $P_\mu^{-\nu}$ is the associated Legendre function (apparently), and $s=\sqrt{p^2-a^2}$, granted that that $p>a$. My problem is that
\begin{align} \frac{p}{s} &= \frac{p}{p\sqrt{1-\frac{a^2}{p^2}}} > 1, \end{align} making the argument of $P_\mu^{-\nu}$ outside the range $[-1,1]$, and thus the value of the function undefined. Did Bateman make a typo? I suspected he meant to write $p/a$ instead of $p/s$, but we still have the same problem.
With \begin{equation} I=\int_0^{\infty} t^\mu I_\nu(at)\,e^{-pt} dt =a^{-\mu-1}\int_0^\infty x^\mu I_\nu(x)e^{-\tfrac{p}{a}x}\,dx \end{equation} the transform exist if $p>a$ and $\Re(\mu+\nu)>-1$. In this paper by Cohl, eq. (11) gives an evaluation of the previous integral from Gradsteyn and Ryzhik (6.624.5) and Prudnikov (2.15.3.2) (with $z=p/a$): \begin{align} \int_0^\infty x^\mu I_\nu(x)e^{-zx}\,dx&=\sqrt{\frac{2}{\pi}}e^{-i\pi\left( \mu+1/2 \right)}\left( z^2-1 \right)^{-\mu/2-1/4}Q_{\nu-1/2}^{\mu+1/2}\\ &=\Gamma(\mu+\nu+1)\left( z^2-1 \right)^{-\mu/2-1/2}P_{\mu}^{-\nu}\left( \frac{z}{\sqrt{z^2-1}} \right) \end{align} for $\Re z>1$ and $\Re(\mu+\nu)>-1$. (Both results are equivalent by Whipple formula). The Laplace integral is then correctly stated: \begin{equation} I=\Gamma(\mu+\nu+1)s^{-\mu-1}P_{\mu}^{-\nu}\left(\frac{p}{s} \right) \end{equation} The Legendre function is unambiguously defined on $(1,\infty)$ (see DLMF, for example).