Balance the cube on a corner, so that there is one corner, $D$ (for Down), touching the table, three corners $A,B,C$ all connected to $D$ with the same height off the table, three corners $X,Y,Z$ with the same height off the table, $A$ connected to $X$ and $Y$, $B$ connected to $Y$ and $Z$, $C$ connected to $Z$ and $X$, and one final corner $U$ (for Up) connected to $X,Y,Z$.
All the spiders start by holding a strategy meeting at $D$. They then split up, and travel until spider $a$ is at $A$, spider $b$ is at $B$, and spider $c$ is at $C$. If they have not yet caught the ant then they make a solemn vow to never let the ant reach any of the edges $AD, BD, CD$ alive, and then they move on to the second phase of their plan.
In phase two, spider $a$ restricts itself to walking back and forth along edge $AX$, spider $b$ along $BY$, spider $c$ along $CZ$. Each spider is trying to catch one of the ant's three "shadows", called shadow $x$, shadow $y$, and shadow $z$.
Let's describe shadow $x$: if the ant is somewhere along the path $AYUX$ of length three, with distance $d$ from $A$ (measured along this path), then shadow $x$ is on line $AX$ with distance $d/3$ from $A$. If the ant is on edge $BY$, then shadow $x$ is on line $AX$ with distance $1/3$ from $A$ (so, the same shadow as when the ant is at $Y$). If the ant is on edge $AX$ or $CX$, shadow $x$ is at $X$ (same shadow as when the ant is at $X$). If the ant is on any of the edges $UZ,BZ,CZ$, then shadow $x$ is on line $AX$ with distance $2/3$ from $A$ (same shadow as when the ant is at $U$). As long as the spiders don't break their vow, then shadow $x$ will always move at most as fast as the spiders, so eventually spider $a$ will catch shadow $x$ and from then on it can move back and forth together with it. In fact, spider $a$ can even guarantee that it catches up to shadow $x$ before shadow $x$ makes it to $A$, guaranteeing that the ant won't make it to edge $AD$ alive.
Shadows $y$ and $z$ are defined similarly, with shadow $y$ along $BY$ according to where the ant is along the path $BZUY$ (or blah blah blah if it is off the path) and shadow $z$ along $CZ$ according to where the ant is along path $CXUZ$ (etc).
Eventually all the spiders successfully catch their corresponding shadows without breaking their vows. If the ant is on any of the edges $AX, BY, CZ$ at this point, then it is already dead. Otherwise, they move on to phase three.
If the ant starts phase three on edge $CX$, then spider $a$ and shadow $x$ will both be at $X$ and spider $c$ and shadow $z$ will be somewhere along $CZ$: spider $c$ now rushes straight back to $C$ before shadow $z$ can get there and then along $CX$ to meet spider $x$, and the ant has no hope of escape. Similarly if the ant starts phase three along edge $AY$ or edge $BZ$ it meets a grisly fate.
If the ant starts phase three on any of the edges $UX, UY, UZ$, then spider $a$ races shadow $x$ to $X$, spider $b$ races shadow $y$ to $Y$, and spider $c$ races shadow $z$ to $Z$. Then all three spiders march straight up to $U$, and the ant is captured.