I believe the following works. It generates all$^1$ candidate lists $\beta_i$ such that $\sum_i i\beta_i' \leq d$ and then compares them element-wise to $\beta$.

generateTuples[beta_, d_] := Module[{lst = PadRight[beta, d], candidates}, candidates = Flatten[FrobeniusSolve[Range[d], #] & /@ Range[d], 1];  TakeWhile[#, # > 0 &] & /@ Select[candidates, And @@ Thread[lst <= #] &] ] 

So:

generateTuples[{1}, 3] (* {{1}, {2}, {1, 1}, {3}} *) 

_{$^1$ I'm not entirey sure that I'm capturing all the possibilities or that I'm understanding the OP's requirement. The OP should check my assumptions and test this with lots of example lists.}

I believe the following works. It generates all$^1$ candidate lists $\beta'$ such that $\sum_i i\beta_i' \leq d$ and then compares them element-wise to $\beta$.

generateTuples[beta_, d_] := Module[{lst = PadRight[beta, d], candidates}, candidates = Flatten[FrobeniusSolve[Range[d], #] & /@ Range[d], 1]; Select[candidates, And @@ Thread[lst <= #] &] /. {a__Integer, 0 ..} :> {a} ] 

So:

generateTuples[{1}, 3] (* {{1}, {2}, {1, 1}, {3}} *) 

_{$^1$ I'm not entirely sure that I'm capturing all the possibilities or that I'm understanding the OP's requirement. The OP should check my assumptions and test this with lots of example lists.}

Explanation

The list $\beta$ is assumed to terminate at the last non-zero entry $i_{\mathrm{max}}$. (This can be fixed to chop off trailing zeroes if necessary.) It is then padded with zeros up to length $d$ so that it can be directly compared with the generated candidates. Since a candidate $\beta'$ must satisfy $\sum_i i\beta_i' \leq d$, and $\beta$ consists of non-negative integers, the longest that $\beta$ can be is of length $d$, since $(0,0,\dots,0,\beta_d = 1)$ makes the sum equal to $d$.

We then generate all lists of non-negative integers satisfying the condition $\sum_i i\beta_i' \leq d$ using FrobeniusSolve, compare each of these candidates to $\beta$ element-wise, and keep those that satisfy $\beta_i \leq \beta_i'$.

(The last little bit /. {a__Integer, 0 ..} :> {a} of code chops off the trailing zeroes in each $\beta_i$.)