Let $(B_t)_{t\geq0}$ be a standard Brownian motion on a probability space $(\Omega,\mathcal{A},P)$ and let $X_t=\exp(B_t-\frac{1}{2}t)$, for $t\geq0$. Let $(\mathcal{F}_t)_{t\geq0}$ with $\mathcal{F}_t=\sigma(B_s\ :\ 0\leq s\leq t)$ be its natural filtration. Further, let $s,t$ be real numbers with $0\lt s\lt t$.
Given $\mathcal{F}_s$, I now want to find a conditional density $f:\Omega\times\Bbb{R}^+\to\Bbb{R}^+_0$ of $X_t$.
If $f$ is a conditional density, then it should be measurable with respect to the Product $\sigma$-algebra $\mathcal{F}_s\otimes\mathcal{B}(\Bbb{R}^+)$ and given $\mathcal{F}_s$, the function $K:\Omega\times\mathcal{B}(\Bbb{R}^+)\to\Bbb{R}^+_0,\ K(\omega, A)=\int_Af(\omega,x)dx$ should be a conditional distribution of $X_t$.
I am unsure how to proceed and would really appreciate any help or solution to this problem, thank you.
The process in your question ($X_t$) is a Geometric Brownian Motion (GBM) with drift $\mu=0$ and diffusion $\sigma=1$, evaluated at $X_0=1$. Its differential equation is the result of applying the Ito-Doeblin formula:
$$ dX_t = X_tdB_t. $$
As a consequence, it is a well known fact that its distribution, conditional to $B$ is a lognormal distribution such as
$$ \ln{X_t}-\ln{X_s}\sim\mathcal{N}(-\frac{1}{2}(t-s),\,\sqrt{t-s}). $$
(Generally a GBM with drift $\mu$ and diffusion $\sigma$ has a differential form
$$ dX_t = \mu X_t dt + \sigma X_t dB_t $$
and integral
$$ X_t = X_s \exp{\{(\mu-\frac{1}{2}\sigma^2)(t-s)+\sigma (B_t-B_s)\}}, $$
which distributes as
$$ \ln{X_t}-\ln{X_s}\sim\mathcal{N}((\mu-\frac{1}{2}\sigma^2)(t-s),\,\,\sigma\sqrt{t-s}). $$)