You can scale up the fonts in your call to sns.set()
.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
x = np.random.normal(size=37)
y = np.random.lognormal(size=37)
# defaults
sns.set()
fig, ax = plt.subplots()
ax.plot(x, y, marker="s", linestyle="none", label="small")
ax.legend(loc="upper left", bbox_to_anchor=(0, 1.1))
sns.set(font_scale=5) # crazy big
fig, ax = plt.subplots()
ax.plot(x, y, marker="s", linestyle="none", label="big")
ax.legend(loc="upper left", bbox_to_anchor=(0, 1.3))