Let \(\Phi\) be an \(N\)-function, \(\Phi(u)=\int_0^{|u|}\varphi(t)\,dt\) where \(\varphi\) is right continuous, \(\varphi(0)=0\), \(\varphi(t)\nearrow\infty\) as \(t\nearrow\infty\). Define the modular \(\rho_\Phi(x)=\sum_{i=1}^\infty \Phi(|x(i)|)\) for sequences \(x=x(i)\). The Orlicz sequence space \(\ell^\Phi\) or \(\ell^{(\Phi)}\) is the class \(\{x(i): \rho_\Phi(\lambda x)0\}\) equipped with the Orlicz norm \(\|x\|_\Phi=\inf_{k>0} k^{-1}[1+\rho_\Phi{kx}]\) or with the Luxemburg norm \(\|x\|_{(\Phi)}=\inf\{\lambda>0: \rho_\Phi(x/\lambda)\leq1\}\), respectively. The normal structure coefficient \(N(X)\) for a Banach space \(X\) has been defined by \textit{W. Bynum} [Pac. J. Math. 86, 427--436 (1980; Zbl 0442.46018)] as \(N(X)=\inf\{d(A)/r(A): A\subset X \text{ bounded with } d(A)>0\}\) where \(r(A)\) is the relative Chebyshev radius of \(A\) with respect to \(co(A)\) and \(d(A)\) is the diameter of \(A\). The main result of the paper establishes exact values for \(N(\ell^\Phi)\) and \(N(\ell^{(\Phi)})\): If the index function \(F_\Phi(t)=t\varphi(t)/\Phi(t)\) is decreasing and \(C^0_\Phi=\lim_{t\to0}F_\Phi(t)>2\), then \(N(\ell^\Phi)=N(\ell^{(\Phi)})=2^{1/C^0_\Phi}\), and if \(F_\Phi(t)=t\varphi(t)/\Phi(t)\) is increasing and \(1