Is a proper ideal of a polynomial ring (on a field) still proper in the polynomial ring on a field extension?

Yes, and this is just linear algebra. Suppose we are looking for polynomials $f_1,\dots,f_k$ of degree at most $m$ such that $\sum f_ip_i=1$. This equation can be expanded out into a system of linear equations in the coefficients of the $f_i$. If this system of linear equations has no solution in $\mathbb{F}$, it also has no solution in any extension of $\mathbb{F}$ (since we can determine whether there is a solution by Gaussian elimination, which uses only the field operations).

Or, from a more sophisticated perspective, let $R=\mathbb{F}[x_1,\dots,x_n]$, $S=\mathbb{K}[x_1,\dots,x_n]$, $I=(p_1,\dots,p_k)\subseteq R$, and $J=(p_1,\dots,p_k)\subseteq S$. We then have canonical isomorphisms $S\cong\mathbb{K}\otimes_\mathbb{F} R$ and $$S/J=S/IS\cong S\otimes_R R/I\cong \mathbb{K}\otimes_\mathbb{F} R/I.$$ If $I$ is proper, then $R/I$ is a nontrivial $\mathbb{F}$-vector space, so $S/J\cong \mathbb{K}\otimes_\mathbb{F} R/I$ is a nontrivial $\mathbb{K}$-vector space, so $J$ is proper.