Thanks to <PERSON> for the link. I think I can prove now that the problem, called PRODUCT, is NP-hard by a reduction from EQUAL-PARTITION, the variant of PARTITION in which we require both parts to have equal sizes.
Given an instance ${x_1,\ldots,x_n}$ of EQUAL-PARTITION (with $x_i > 0$ and $n$ is even), consider an instance of my problem with $k = n/2$, $\Delta=\left(1+nM/2+A/2\right)^2$ for large enough $M$ (i.e., $M = \max_i x_i$ ) and $A=\sum_{i}x_i$ , and let $a_i=M+x_i$ for all $i$ and $b_i=M+2A/n-x_i$ for all $i$.
This is a polynomial time reduction.
Now let us prove that EQUAL-PARTITION is solved $\iff$ PRODUCT is solved.
* "$\Rightarrow$" Suppose that EQUAL-PARTITION is solved. Then there exists $S$ of size $|S|=n/2$ such that $$ \sum_{i\in S}x_i=\sum_{i\in S'}x_i=A/2, $$ where $S\cup S'={1,\ldots,n}$ and $S\cap S'=\emptyset$ and $|S|=|S'|=n/2$. 262
Now take the solution to PRODUCT to be $S$. We have \begin{align} &\left(1+\sum_{i\in S}a_i\right)\left(1+\sum_{i\in S}b_i\right)\geq\Delta\ \iff&\left(1+\sum_{i\in S}(M+x_i)\right)\left(1+\sum_{i\in S}(M+2A/n-x_i)\right)\geq\Delta\ \iff&\left(1+nM/2+\sum_{i\in S}x_i\right)\left(1+nM/2+A-\sum_{i\in S}x_i\right)\geq\Delta\ \iff&\left(1+nM/2+A/2\right)\left(1+nM/2+A/2\right)=\left(1+nM/2+A/2\right)^2\ \end{align}
* "$\Leftarrow$" Suppose that PRODUCT is solved. Then there exists $S$ of size $|S|=n/2$ such that \begin{align} \require{overset} &\left(1+\sum_{i\in S}a_i\right)\left(1+\sum_{i\in S}b_i\right)\geq\Delta\ \iff&\left(1+\sum_{i\in S}(M+x_i)\right)\left(1+\sum_{i\in S}(M+2A/n-x_i)\right)\geq\Delta\ \iff&\left(1+nM/2+\sum_{i\in S}x_i\right)\left(1+nM/2+A-\sum_{i\in S}x_i\right)\geq\Delta\ \iff&\left(\alpha+\sum_{i\in S}x_i\right)\left(\alpha+A-\sum_{i\in S}x_i\right)\geq\Delta\ \iff&\alpha^2+\alpha A+A\sum_{i\in S}x_i-\left(\sum_{i\in S}x_i\right)^2\geq\Delta\ \iff&\left(\sum_{i\in S}x_i\right)^2-A\sum_{i\in S}x_i+\Delta-\alpha^2-\alpha A\leq 0\ \iff&X^2-AX+\Delta-\alpha^2-\alpha A\leq 0\ \overset{(a)}{\iff}&X^2-AX+\Delta-\alpha^2-\alpha A= 0\ \iff&X= A/2\ \end{align}
Finally, PRODUCT is NP-hard. 262