Let me formalize the problem and give some results.

A number $n$ is $y$-smooth (or $y$-friable) if all its prime factors $p$ are less than or equal to $y$. Let us set
$$\Psi(x,y) :\# \{ n \le x: n \text{ is }y\text{-smooth}\},$$
which appeared in Sungjin Kim's comment. One can consider the quantity
$$S_k(x,y) := \# \{ n_1,\ldots,n_k \le x: \gcd(n_1,n_2,\ldots,n_k)=1, \,  n_i \text{ is }y\text{-smooth}\}.$$ 
The ratio $S_k(x,y)/\Psi(x,y)^k$ is exactly the probability that $n$ positive integers chosen uniformly at random from the set of $y$-smooth numbers up to $x$ have no (joint) non-trivial common factor. 

If $n \le x$ then $n$ is $x$-smooth, so $S_k(x,x)/\Psi(x,x)^k = S_k(x,x)/\lfloor x \rfloor^k$ is the probability that $n$ positive integers $\le x$ chosen uniformly at random have no common factor, and this converges to $1/\zeta(k)$. You seem to ask about $S_k(x,y)/\Psi(x,y)^k$ when $y$ is fixed.

We can use inclusion-exclusion to write $S_k(x,y)$ as
$$\begin{align} S_k(x,y) &= \sum_{d \mid \prod_{p \le y}}\mu(d) \# \{ m_1,\ldots,m_k \le x/d: m_i \text{ is }y\text{-smooth}\}\\ &=\sum_{d\text{ is }y\text{-smooth}} \mu(d)\Psi(x/d,y)^k\end{align}$$
where $d$ stands for the common divisor of $k$ $y$-smooth numbers $n_1,\ldots,n_k$. So we need to understand
$$ \frac{ \sum_{d\text{ is }y\text{-smooth}} \mu(d)\Psi(x/d,y)^k}{\Psi(x,y)^k}.$$

Ennola (1969) proved that $\Psi(x,y) \sim C_y (\log x)^{\pi(y)}(1+O_y(1/\log x))$ where $\pi(y)$ is the number of primes up to $y$ and $C_y$ is an explicit positive constant. Since $\log x \sim \log x/d$ for any fixed $d$, and since $\sum_{d\text{ is }y\text{-smooth}}\mu(d)=0$ for $y \ge 2$, this immediately implies that the above fraction is $O_y(1/\log x)$, so the probability tends to $0$. So most (in a limit sense) $k$-tuples of $y$-smooth integers \textbf{have} a common factor, as observed numerically by Stefan Kohl and Aaron Meyerowitz

If you allow $y$ to grow with $x$, then much more can be said, because the ratio $\Psi(x/d,y)/\Psi(x,y)$ was studied extensively in the literature, namely in 

 1. Adolf Hildebrand and Gérald Tenenbaum, "On integers free of large prime factors". Trans. Amer. Math. Soc. 296 (1986), no. 1, 265–290.
 2. Aleksandar Ivić and Gérald Tenenbaum, "Local densities over integers free of large prime factors". Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 148, 401–417.
 3. Régis de la Bretèche and Gérald Tenenbaum, "Propriétés statistiques des entiers friables", Ramanujan J. 9 (2005), no. 1-2, 139–202.
 4. See also Theorem III.5.22 in Tenenbaum's book (3rd edition of English version).

Roughly speaking, in Theorem 3 of the Hildebrand--Tenenbaum paper, they show in certain ranges of parameters, $\Psi(x/d)/\Psi(x,y) \sim d^{-\alpha}$ (with reasonable error term) where $\alpha$ is a parameter depending only on $x$ and $y$ (arising as a saddle point of some generating function); $\alpha$ is studied in Theorem 2(i) and its proof. If $y\gg \log x$ it is close to $1-(\log \log x/\log y)$ (see also the Ivić--Tenenbaum for more precise results in the range $y \gg \log x$), while for small $y$ it is close to $y/(\log x \log y)$.

*Carefully* using these results should possibly lead you to a result of the shape
$$ \frac{ \sum_{d\text{ is }y\text{-smooth}} \mu(d)\Psi(x/d,y)^k}{\Psi(x,y)^k} \sim \sum_{d\text{ is }y\text{-smooth}} \mu(d)d^{-k\alpha} = \prod_{p \le y} \left(1-p^{-k\alpha}\right)$$
for some range of $y$ and $x$, where $\alpha$ is the above-mentioned constant. If, say, $y=(\log x)^C$ for some $C \ge 1+\varepsilon$, this $\alpha$ is asymptotic to $1-1/C$. If $y$ is at least a power of $x$, say, then $\alpha$ is asymptotic to $1$ (in fact, very close to $1$) and the final result is potentially asymptotic to $1/\zeta(k)$ (I haven't checked).

I'll end by mentioning that in the La Bretèche--Tenenbaum paper they study the number of $y$-smooth numbers up to $x$ that are coprime to a given integer $m$, obtaining results uniform in $m$. This can be expressed as $\sum_{d \mid m}\mu(d) \Psi(x/d,y)$, which is similar to the quantity mentioned above, so maybe the best results are obtained using the ideas in that work. In particular, they show that $$\Psi(x/d,y) \ll \Psi(x,y)d^{-\alpha}$$
uniformly for $x \ge y \ge 2$, $1 \le d \le x$, which is powerful when trying to truncate the sum over $d$.