The line with equation $y = x$ has slope 1 and passes through $(0,0)$. When this line is translated, its slope does not change. When this line is translated 3 units to the right and 2 units down, every point on the line is translated 3 units to the right and 2 units down. Thus, the point $(0,0)$ moves to $(3,-2)$. Therefore, the new line has slope 1 and passes through $(3,-2)$. Thus, its equation is $y - (-2) = 1(x - 3)$ or $y + 2 = x - 3$ or $y = x - 5$. The $y$-intercept of this line is -5.