Actually no, I asked this question as a follow-up to #138581, which I only saw today. I have to admit that I do not even know the density of $q \in \mathbb{Q}$ for which $qy^2 = x^2 +1$ has a solution... (although this should be easy). I will think about this and other examples. But I was motivated by this basic observation: if we let $q$ run through the primes alone, then the density is $1/2$.

@FelipeVoloch: I just saw your paper with B. Poonen, in which you exclude the supposedly trivial case $d = n = 2$ of plane conics. Is the density zero in that case?

Thanks for spotting this, I will edit. I had considered the curves to be affine in relation to the other problem (in genus $> 1$, where the density should be $0$ unless there is a section); its purpose there was to remove a finite number of sections from a family of projective curves. But as you write, this has no significance in the genus zero case considered here.

About the issue of letting $g \to +\infty$: For each individual $g > 1$, surely the density of pointless hyperelliptic curves of genus $g$ (denoted $\rho_g = 1-o(2^{-g})$ in Bhargava's paper) should equal $1$. Wouldn't this imply, a fortiori, the statement "most hyperelliptic curves are pointless," taken in any reasonable sense? This refers to your comment following JSE's answer.

I just saw this. I had also asked this question at mathoverflow.net/questions/121104/… . To state the obvious, in any flat family $X \to B$ of geometrically irreducible affine curves over $\mathbb{Q}$ (over an integral base $B$), one of the following alternatives should hold: (1) there is a section to $X \to B$; or (2) the pointless curves in the family have a full density (as in JSE's answer below); or (3) the geometric genus of the generic member is $\leq 1$. One could also ask about the possible densities for the genera $0$ and $1$.

I just saw this. I had asked the exact same question ("either every curve in the family X has a point, or almost none of them do") in mathoverflow.net/questions/121104/… .

Would your question be answered by Lemma 5.3 in J. S. Milne's text on abelian varieties, and the discussion preceding it? Here is the link: jmilne.org/math/CourseNotes/AV.pdf