\documentclass{article}

\usepackage{amsmath,amssymb}

\newcommand\setst[2]{\{#1 :
  \begin{array}[t]{@{}l@{}}#2 \}\end{array}}

% my attempted mod
\newcommand\Setst[3][0pt]{%
  \expandafter\setsttopandbottom#3\relax
  \begin{array}{@{}l @{\ } l@{}}
    \left\{#2 :\vphantom{\setsttop}\vphantom{\setstbottom}\right.\kern-\nulldelimiterspace &
    \begin{array}[t]{@{}l@{}}
      \setsttop
      \\[#1]
      \kern-\nulldelimiterspace\left.\setstbottom\vphantom{\setsttop}\right\}
    \end{array}
  \end{array}
  }

\def\setsttopandbottom#1\\#2\relax{\def\setsttop{#1}\def\setstbottom{#2}}

\begin{document}

OK:
\[
  \setst{f}{f(t) > 0 \text{ for all $t$ with } 0 < t < 1 \\
      \text{and }\left|\int_{0}^{1}e^{-\frac{1}{2} t^{2}}f(t) dt\right| \leq 1 }
\]

Modified:
\[
  \Setst[20pt]{f}{f(t) > 0 \text{ for all $t$ with } 0 < t < 1\\
      \displaystyle\text{and }\left|\int_{0}^{1}e^{-\frac{1}{2} t^{2}}f(t) dt\right| \leq 1 }
\]

\end{document}